Thermomechanics of solids: Nonequilibrium and irreversibility

Reconciliation of thermal and mechanical interdependence is made possible by casting away the concept of heat. Defined are the dissipation and available energy density as two mutually exclusive quantities; thermal and mechanical changes become two aspects of the same process that coexist and are in continual operation. This is accomplished by interlacing the rotation, deformation and change in element size into one single operation; irreversibility is embedded inherently into the theory. No a priori assumption needs to be made on the constitutive relations which are, in fact, derived for each individual element and time increment. Only the initial slope of the reference material and load history need to be specified. Instead, the surface and volume energy density are assumed to be exchangeable without letting the change of volume with surface area to vanish in the limit, a simplification of classical physics and continuum mechanics that results in the decoupling of thermal and mechanical effects. Complete nonlinearity and finiteness of deformation are retained such that boundary problems can be solved directly by specifying the tractions and/or displacements. Nonequilibrium/irreversible solutions are shown to possess definite limits and to be bounded by the equilibrium/irreversible states whose solutions are proved to be unique. The existence of the isoenergy density function provides an elegant means of resolving the multidimensionality of the problem; the translation of unidimensional data to multidimensional states.     

Micro/macro-crack growth due to creep–fatigue dependency on time–temperature material behavior

The implicit character of micro-structural degradation is determined by specifying the time history of crack growth caused by creep–fatigue interaction at high temperature. A dual scale micro/macro-equivalent crack growth model is used to illustrate the underlying principle of multiscaling which can be applied equally well to nano/micro. A series of dual scale models can be connected to formulate triple or quadruple scale models. Temperature and time-dependent thermo-mechanical material properties are developed to dictate the design time history of creep–fatigue cracking that can serve as the master curve for health monitoring.In contrast to the conventional procedure of problem/solution approach by specifying the time- and temperature-dependent material properties as a priori, the desired solution is then defined for a class of anticipated loadings. A scheme for matching the loading history with the damage evolution is then obtained. The results depend on the initial crack size and the extent of creep in proportion to fatigue damage. The path dependent nature of damage is demonstrated by showing the range of the pertinent parameters that control the final destruction of the material. A possible scenario of 20 yr of life span for the 38Cr2Mo2VA ultra-high strength steel is used to develop the evolution of the micro-structural degradation. Three micro/macro-parameters μ^*, d^* and σ^* are used to exhibit the time-dependent variation of the material, geometry and load effects. They are necessary to reflect the scale transitory behavior of creep–fatigue damage. Once the algorithm is developed, the material can be tailor made to match the behavior. That is a different life span of the same material would alter the time behavior of μ^*, d^* and σ^* and hence the micro-structural degradation history. The one-to-one correspondence of the material micro-structure degradation history with that of damage by cracking is the essence of path dependency. Numerical results and graphs are obtained to demonstrate how the inherently implicit material micro-structure parameters can be evaluated from the uniaxial bulk material properties at the macroscopic scale.The combined behavior of creep and fatigue can be exhibited by specifying the parameter ξ with reference to the initial defect size a₀. Large ξ (0.90 and 0.85) gives critical crack size a_cr = 11–14 mm (at t < 20 yr) for a₀ about 1.3 mm. For small ξ (0.05 and 0.15), there results critical a_cr = 6–7 mm (at t < 20 yr) for a₀ about 0.7–0.8 mm. The initial crack is estimated to increase its length by an order of magnitude before triggering global to the instability. This also applies ξ ≈ 0.5 where creep interacts severely with fatigue. Fine tuning of a_cr and a₀ can be made to meet the condition oft = 20 yr.Trade off among load, material and geometric parameters are quantified such that the optimum conditions can be determined for the desired life qualified by the initial–final defect sizes. The scenario assumed in this work is indicative of the capability of the methodology. The initial–final defect sizes can be varied by re-designing the time–temperature material specifications. To reiterate, the uniqueness of solution requires the end result to match with the initial conditions for a given problem. This basic requirement has been accomplished by the dual scale micro/macro-crack growth model for creep and fatigue.     

Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics

Any attempt made to separate energy into electrical and mechanical parts may lead to inconsistencies as they do not necessarily decouple. This is illustrated by application of the energy density function in the linear theory of piezoelasticity. By assuming that a critical energy density function prevails at the onset of crack initiation, it is possible to establish the relative size of an inner and outer damage zone around the crack tip; they correspond to the ligaments at failure caused by pure electric field and pure mechanical load. On physical grounds, the relative size of these zones must depend on the relative magnitude of the mechanical and electrical load. Hence, they can vary in size depending on the electromechanical material and damage resistance properties. Numerical results are obtained for the PZT-4, PZT-5H, and P-7 piezoelectric ceramics. These two ligaments for the two damage zones may coincide for appropriate values of the applied electrical field and mechanical load.Explicit expression of the energy density factor S is derived showing the mixed mode electromechanical coupling effects. The factor S can increase or decrease depending on the direction of the applied electric field with reference to the poling direction. This is in contrast to the result obtained from the energy release rate quantity, which remains unchanged for electric field in the direction of poling or against it.     

Irreversibility and damage of SAFC-40R steel specimen in uniaxial tension

Macroscopic material damage is detected and assessed for the SAFC-40R steel specimen in uniaxial tension even when the stress responded linearly with strain. As the loading increased monotonically at a rate of 0.2 cm/min, the specimen first absorbed heat from the surrounding and then released heat when the strain is almost five times beyond the so-called “elastic limit”. In other words, the specimen undergoes cooling and heating with reference to the ambient temperature. This phenomenon is predicted theoretically for the first time by application of the energy density theory and the results agreed well with experimental data. Obtained is the H-function that possesses a distinct threshold at time between 21 and 22 seconds after loading. This transition is defined as the onset of disorder at which point the energy dissipation density D increases suddenly by one order of magnitude. The corresponding uniaxial stress and strain are 194.4 MPa and 0.9764·10⁻³ cm/cm, respectively. These values are lower than those normally referred to at the yield point.     

Directional dissimilarity of transitional functions: Volume energy density factor

Scale segmentation of the SI system of measurement has necessitated the use of transitional function for connecting the results of the segmented scales. The state-of-the-art of measurement in the 1800s is reflected by the coarseness of scale segmentation. By up-to-day standard of nanonization, scale refinement is necessary to account for the interactive benefits of multiscaling, the building block of which is dual scaling, say macro-micro or micro-nano. The directional dissimilarity between two adjacent scales, say macro → micro and micro → macro, arises naturally in physical processes. An index Λ is defined to measure the severity of directional dissimilarity that may prevail in the volume energy density factor (VEDF) for macro → micro. A similar index Ω applies to micro → macro. Both A and Ω reflect the combined effects of loading, material and geometry. The macro-micro transition of VEDF_macro can differ from the micro-macro transition of VEDF_micro. The relation between the macrostress intensity factor and VEDF_macro is incompatible with that for the microstress intensity factor and VEDF_micro. The stress intensity factor, a monoscale concept, will not hold for both the macro- and micro-scale. To begin with, the units may differ. The same holds for the volume energy density factor. The irreconcilability can be resolved by an argument of dimensional compatibility, provided that the agreement is to preserve the monoscale definitions. These considerations call for a scrutiny of the crack growth data whose scatter may be contributed by the directional effects of scale transition.     

Multiscale reliability of physical systems based on the principle of least variance

Multiscale reliability places priority on the shifting of space-time scale while dual-scale reliability concentrates on time limits. Both can be ranked by applying the principle of least variance, although the prevailing criteria for assessment may differ. The elements measuring reliability can be ideally assumed to be non-interactive or interactive as a rule. Different formulations of the latter can be adopted to yield weak, strong, and mixed reliability depending on the application. Variance can also be referred to the average based on the linear sum, the root mean square, or otherwise. Preference will again depend on the physical system under consideration. Different space-time scale ranges can be chosen for the appropriate time span to failure. Up to now, only partial validation can be made due to the lack of lower scale data that are generated theoretically.A set of R-integrals is defined to account for the evolution effects by way of the root functions from Ideomechanics. The approach calls for a “pulsating mass” model that can connect the physical laws for the small and large bodies, including energy dissipation at all scale level. Non-linearity is no longer an issue when characterization of matter is made by the multiscaling of space-time. Ordinary functions can also be treated with minor modifications.The key objective is not to derive new theories, but to explain the underlying physics of existing test data, and the reliability of diversified propositions for predicting the time span to failure. Present and past investigations have remained at the micro-macro or mi-ma scale range for several decades due to the inability to quantify lower scale data. To this end, the available mi-ma fatigue crack growth data are used to generate those at the na-mi and pi-na scale ranges. Reliability variances are computed for the three different scale ranges, covering effects from the atomic to the macroscopic scale. They include the initial crack or defect length and velocities. Specimen with large initial defects are found to be more reliable. This trend also holds for each of the na-mi and pi-na scale range. Also, large specimen data had smaller reliability variances than the smaller specimens making them more reliable. Variances for the nano- and pico-scale range had much more scatter and were diversified. Uncertainties and un-reliabilities at the atomic and sub-atomic scale are no doubt related, although their connections remain to be found.Reliability with high order precisions are also defined for multi-component systems that can involve trillions of elements at the different scale ranges. Such large scale computations are now within reach by the advent of super-speed computers, especially when reliability, risk, and among other factors may have to be considered simultaneously.     

Edge dislocations generated from a microcrack under initial residual stress of non-uniform distribution

Crystal nucleation gives rise to inhomogeneity in the crystal lattice. The prevailing stresses and strains caused by non-uniform cooling can create microcracks with residual stresses locked-in at the end segments. These stresses can have a non-uniform distribution where the amplitude can increase or decrease from the microcrack tip which is highly strained to generate edge dislocations under in-plane shear. A dual scale microdislocation crack model is considered by focusing attention near the microcrack tip singularity such that more than 10 orders of magnitude in lineal dimension can be covered from the atomic to the microscopic scale. The concept of a scale multiplier is employed to connect the microscopic and atomic scale results. Discontinuity at the cross-scaling location is necessitated by dividing the full range of the non-equilibrium process into two regions within which equilibrium mechanics can be used. When needed, additional mesoregions can be added to reduce the transient discontinuities.Solved in closed form is the solution for the generation of edge dislocations due to non-uniform residual stress distributions at the end segment of the microcrack tip which will henceforth be referred to simply as the “tip”. Three different Cases I, II and III will be considered where the residual stress will possess a peak at the different locations. Case I for the furthest away from the tip, Case II for the peak nearest to the tip and Case III for the peak in the middle of the residual stress segment. Compared are the scale multiplier α whose maximum value being one corresponding to no discontinuity at cross-scaling. Hence, small α corresponds to large discontinuity. For Cases I, II and III, αs are found, respectively as 0.17, 0.43 and 0.28. The largest discontinuity occurred at α = 0.17 when the peak of the residual stress is farthest away from the microcrack tip. The largest number of edge dislocations or imperfections are also generated for Case I. The precise location of the residual stress peak is related to the magnitude and the segment length of the residual stress. These findings are manifestation of the variety of non-homogeneities that can arise in a metal alloy during crystal formation, not to mention the prevailing conditions at the grain boundaries. The idea is not to account for the details per se but to test the sensitivities of the microscopic and atomic parameters involved. To this end, the energy density function for the dual scale model will be determined and discussed in connection with what has been emphasized.     

Impact damage of laminated composite with energy dissipation: Part II — Damage evolution

Having developed the methodology for analyzing the failure of a ceramic/rubber/steel composite laminate impacted by a tungsten rod in Part I, Part II of the work is concerned with the progressive damage process where material continuity would be interrupted at different locations and time intervals. Depending on the time rate dependent threshold values of the surface and volume energy density, the degree and extent of damage by fragmentation, mass loss, etc. are determined by finite element calculations for time steps of 0.15, 5.0, 7.5, 10, 20, 21 and 21.5 μs. Stresses and strains possess an oscillatory character in time; they alternate in sign as the impact waves bounce back and forth in the three-layered dissimilar materials.Local strain rates of approximately 10⁵, 10³ and 10⁴ s⁻¹ are formed in the ceramic, rubber and steel layer respectively at locations underneath the tungsten rod after 16 μs of impact. A more wide range of strain ratio would have prevailed for a homogeneous layer of the same thickness. The tungsten rod is now badly fragmented while cracking near the surface of the ceramic is also predicted. Local temperature and dissipation energy density rise rapidly as time approached 20 μs. The maximum surface and volume.energy density in the ceramic near the impact region reached 260 MPa · m and 6.39 MPa, respectively. Complete disintegration of the tungsten rods occurred at 21.5 μs. At this time, the ceramic layer is perforated and the rubber layer is partially cracked. The back-up steel plate, however, remained in tack. These predictions agree qualitatively with past observations.     

Energy dissipation in highly compressed cylindrical bar specimens

Energy dissipated in a highly compressed 4340 steel cylinder is determined as a function of load time history. Analyzed in particular are the different material damage modes. Highly localized deformation can result in phase transformation of the metal in addition to other forms of failure such as yielding and fracture. Under large deformation, material response depends not only on strain rate but also the degree of mechanical damage. These effects are assessed quantitatively by an energy balance approach and the results compare favorably with available experimental data. The energy transfer associated with the change in material microstructure due to phase transformation is calculated. This can be identified with localized deformation bands which have also been observed in explosively fragmented bodies.