A new method of resolving transient stresses in dynamic holophotoelasticity

A method which allows the simultaneous separation of the isochromatic and isopachic fringes for transient plane-stress problems is presented. A double pulsed ruby laser and a terbium glass Faraday rotator is employed to resolve the transient isochromatic and isopachic patterns. Separated fringe patterns for a structural component are recorded at nine different instants after impact loading. The dynamic material-fringe values of isochromatics and isopachics are obtained within the experiment. Finally, dynamic stresses distributed along a section at different time intervals are resolved.     

Transient responses of beams and plates subject to travelling load. Miscellaneous results

The steady-state response of a free and infinite Timoshenko beam is specified analytically in terms of non-dimensional displacements and stresses. The beam is supposed loaded by a travelling concentrated force or a moving step load. By a validated explicit numerical calculation, it is shown how a load travelling on a beam at constant velocity, from defined time and abscissa, generates a response which evolves towards the steady-state solution for a part, and towards a quantified transient solution for another part. Asymptotic values are given for the transient displacements and stresses according to the time and the speed of the loading. The solution is also found for a plate subject to a pressure, which spreads respecting the cylindrical symmetry. It is possible to identify in the response a part which follows the pressure front, and which is comparable with the steady-state response of a beam, and another transient part, which generates displacements and stresses with a much less oscillating character. An asymptotic solution is also presented for the plate.The whole series of the results makes it possible to better understand qualitatively the beginning of the transient response of a beam or of a plate to a moving load, and also makes it possible to estimate the stresses and displacements without needing specialised numerical codes.     

Magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder

In the present paper, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses induced by a transient magnetic field in a conducting solid circular cylinder are investigated. It is assumed that a transient magnetic field which is defined by an arbitrary function of time acts on the surface of the solid cylinder in the direction parallel to its surface. Fundamental equations of plane axisymmetrical electromagnetic, temperature, and elastic fields are formulated. Then, solutions of magnetic field, eddy current, temperature change and both dynamic solutions and quasi-static ones of stresses and deformations are analytically derived in the forms including the arbitrary function. The solutions of stresses are determined to be sums of thermal stress caused by eddy current loss and magnetic stress caused by Lorentz force. For this case that the arbitrary function is given by the smoothed ramp function with sine function, the dynamic and quasi-static behaviors of the stresses are examined by numerical calculations.     

A generalized method for photoelastic studies of transient thermal stresses

This paper describes the apparatus and experimental method which was developed for generalized studies of transient thermal stresses in photoelastic models of many different shapes under a variety of steady-state or transient temperature conditions. It explains how the desired temperature gradients are established in the models and how rapidly changing temperature and stress profiles are monitored during a test. The experimental method is used to study the stresses in a three-dimensional photothermoelastic model subjected to three different temperature sequences. These are: symmetrical cooling of both faces of a thick plate initially at a uniform temperature; heating of one face only of a thick plate initially at a uniform temperature; and heating of only the cold face of a thick plate with an initial linear temperature gradient through its thickness. The last sequence generated temperature profiles which relate to conditions where internal heating is present. The resultant temperature and stress histories for each case are presented graphically and similarity scales are applied to give correct time-stress relations for a typical steel prototype. The magnitude and time of occurrence of the peak stresses on the boundary, as well as in the interior of the plate are found. These stresses are very high and occur comparatively late in each test, at a time when the temperature of the central plane has already started to respond to the changing conditions at the surface. The model was of the sandwich-type construction used by previous investigators, which has a built-in polariscope to isolate a transverse plane for viewing.     

TRANSIENT THERMAL RESPONSE IN THICK ORTHOTROPIC HOLLOW CYLINDERS WITH FINITE LENGTH： HIGH ORDER SHELL THEORY

The transient thermal response of a thick orthotropic hollow cylinder with finite length is studied by a high order shell theory. The radial and axial displacements are assumed to have quadratic and cubic variations through the thickness, respectively. It is important that the radial stress is approximated by a cubic expansion satisfying the boundary conditions at the inner and outer surfaces, and the corresponding strain should be least-squares compatible with the strain derived from the strain-displacement relation. The equations of motion are derived from the integration of the equilibrium equations of stresses, which are solved by precise integration method （PIM）. Numerical results are.obtained, and compared with FE simulations and dynamic thermo-elasticity solutions, which indicates that the high order shell theory is capable of predicting the transient thermal response of an orthotropic （or isotropic） thick hollow cylinder efficiently, and for the detonation tube of actual pulse detonation engines （PDE） heated continuously, the thermal stresses will become too large to be neglected, which are not like those in the one time experiments with very short time.     

Separation of principal stresses in dynamic photoelasticity

A new and effective method used to separate the transient principal stresses for dynamic photoelasticity is proposed. This is a hybrid method combining the optical method of dynamic caustics and the boundary element numerical method. Firstly, a modified Cranz-Schardin spark camera is used to record simultaneously the isochromatic fringe patterns of photoelasticity and the shadow spot patterns in the dynamic process. By means of the isochromatic fringe patterns, the difference between transient principal stresses in the whole domain and the principal stresses along the free boundary can be solved. In addition, the method of caustics is a very powerful technique for measuring the concentrative load. Then, the sum of the principal stresses is calculated by the boundary integral equation obtained from the Laplace integral transform of the wave equation. So, the transient principal stresses can be determined from the experimental and numerical results. As an example, the transient principal stresses in a polycarbonate disk under an impact load are resolved. Concurrently published in the Chinese Edition of Acta Mechanica Sinica, Vol. 26, No. 1, 1994     

有限长厚壁管在过冷沸腾状态下的热应力响应

在求得有限长厚壁管水淬时瞬态温度分布的基础上［１］，引入了包含相变的热弹塑性本构方程的增量形式．用有限元法求得了瞬态热应力和残余应力．对影响热应力和残余应力的各种因素进行了分析和讨论．     

Damage of hybrid composites under long term hygrothermal loading and stacking sequence

The exposure of polymeric composite matrix to a cyclic moist environment produces transient residual stresses extremely important at the edges of the laminated plates, particularly at first times. In cases of critical cyclic environmental conditions, the damage of composites becomes very probable, so durability is intensively reduced. To avoid the damage probability, and to reduce the transient hygrothermal stresses, in our paper the hybrid composites with optimal stacking sequences are used. The first aim of this paper is the determination of hygrothermal characteristics for balanced stacking sequences [θ/−θ]_nS. Therefore, in order to locate the favourite directions of each stacking sequence, the polar representation method is adopted. The suitable choice of hygrothermal characteristics, allows thereafter, the reduction of transient hygrothermal stresses. By using the quadratic failure criterion in stress space, the transient strength ratio is evaluated at first ply of the hybrid plate for each sequences [θ/−θ]_nS from initial time until saturation time. Avoid the damage probability is reached by the reduction of transient hygrothermal stresses along the thickness of the hybrid plate. To locate the minimal and maximal stresses, a progressive variation of the relationship between the thicknesses h₁ and h₂ of hybrid composites constituents AS3501 and T300/5208 was carried out. In this way five cases are proposed, on which, the thickness of each material varies gradually, but the total thickness of our plate remains constant. This procedure permits us to find the best configuration which will offer favourable conditions of services, i.e., to predict a considerable reduction of hygrothermal transverse stresses at both edges of the hybrid plate.     

动态光弹性方法的主应力分离的研究

本文提出了动态光弹性、动态焦散线实验方法同边界元法结合的混合法，并用这种方法解决了动态光弹性主应力分离的问题，首先对现有的多火花高速摄影系统进行了改造，在动态实验过程中，成功地得到了不同瞬时的清晰的动态光弹性的等差线条纹和动态焦散线系列图像，这样，便可提取不同瞬时的边界应力值、全场主应力差值及边界上的外载。继而提出用Ｌａｐｌａｃｅ变换域上的边界元法来求解在冲击载荷作用下二维弹性模型全场的主应力和。最后，以受冲击载荷作用的圆盘为例，进行实验及边界元法计算，得到了分离的主应力场。     

Full field analysis of an anti-plane interface crack subjected to dynamic body forces

In this study, the transient full field response of an interface crack between two different media subjected to dynamic body force at one material is investigated. For time t < 0, the bimaterial medium is stress free and at rest. At t = 0, a concentrated anti-plane dynamic point loading is applied at the medium as shown in Fig. 1. The total wave field is due to the effect of this point loading and the scattering of the incident waves by the interface crack. An alternative methodology that is different from the conventional superposition method is used to construct the reflected, refracted and diffracted wave fields. A useful fundamental solution is proposed in this study and the full field solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying an exponentially distributed traction (in the Laplace transform domain) on the interfacial crack faces. The Cagniard–de Hoop method of Laplace inversion is used to obtain the transient solution in time domain. Exact transient closed form solutions for stresses and stress intensity factors are obtained. Numerical results for the time history of stresses and stress intensity factors during the transient process are discussed in detail.     

Transient thermal stresses in a strip with an eccentric hole

A procedure is presented for dealing with the constraint associated with the direction of the thickness of a thin model and for obtaining transient thermal stresses under plane-stress conditions; the stresses are induced by severe thermal loads. Thermal-stress-concentration factor in an unrestrained strip with an eccentric circular hole placed near the thermally loaded edge is obtained. One straight edge of the strip is cooled and the others are insulated. Consequently, two cases for which the hole acts remarkably as a stress raiser are found.     

A non-linear viscoelastic model with structure-dependent relaxation times: II. Comparison with l.d. polyethylene transient stress results

A model of non-linear viscoelasticity with relaxation times which depend on the structure is compared with experimental results reported in the literature for a L.D. polyethylene.The single parameter of the model is determined by comparison with steadystate shear results. The model is then used to interpret various transient data. These are: tangential and normal stress growth in shear, stress growth in elongation, normal stresses in shear creep.The comparison shows a good general agreement, thus supporting the suggestion of relating the change in time of the relaxation spectrum to structural variables.     

Thermoelastic transient response of an infinitely long annular multilayered cylinder

Thermoelastic transient response of multilayered annular cylinders of infinite lengths subjected to known temperature at traction-free inner and outer surfaces are considered. A method based on the Laplace transformation and finite difference method has been developed to analyze the thermoelasticity problem. Using the Laplace transform with respect to time, the general solutions of the governing equation are obtained in transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. Solutions for the temperature and thermal stress distributions in a transient state were obtained. It was found that the temperature distribution, the displacement and the thermal stresses change slightly as time increases. There is no limit of number of annular layers of the cylinder in the presented computational procedures.     

Theoretical analysis,numerical calculation and experimental measurement of transient elastic waves in strips subjected to dynamic loadings

In this study, the transient response of an elastic strip subjected to dynamic in-plane loadings on the surface is investigated in detail. One of the objectives of this study is to develop an effective analytical method for determining transient solutions in a strip. By applying Laplace transform, the analytical solution in the transformed domain is derived and expressed in matrix form. The solution is then decomposed into infinite wave groups in which the multiple reflected waves with the same reflection are involved. Each multi-reflected wave can be identified by a coding method and be verified by the theory of generalized ray. The inverse transform is performed by using the well-known Cagniard method. The transient solutions in time domain for stresses and displacements are expressed in a closed form and are discussed in detail by an example. The experimental results show that the early time transient responses of displacements on the surface agree very well with the numerical calculations based on the theoretical solutions.     

The role of transient rheology in polymeric sintering

The initial theory of Frenkel and Eshelby for the coalescence of drops in air (or sintering) of Newtonian fluids, which equated the work of surface tension to the work done by viscous stresses while assuming biaxial extensional flow kinematics, was extended to the case of time-dependent material functions using the Upper Convected Maxwell (UCM) model. A numerical scheme was developed to solve the ordinary differential equations (ODE) for the stresses, which are embedded in the ODE based on the mechanical energy balance. Initial conditions required to solve the set of non-linear ODEs were obtained from visualization experiments of the coalescing drops as the theory for elastic contact gave unrealistically high values of the initial neck radius. The transient model predicted that coalescence was accelerated by increasing the relaxation time, the opposite relationship of what was predicted by the steady-state UCM formulation, and was capable of quantitatively predicting the experimental coalescence rates at times when viscoelasticity was important.     

Transient dynamic analysis of interface cracks in anisotropic bimaterials by the scaled boundary finite-element method

The transient response of finite bimaterial plates with interface cracks is analyzed directly in the time domain by using the scaled boundary finite-element method. A bimaterial plate is divided into a few subdomains. Only the boundaries of the subdomains are discretized with line elements leading to great flexibility in mesh generation. The displacement and stress fields are expressed as a series solution which separates the singular stress term from other high-order terms. The oscillatory stress singularity in the radial direction emanating from the scaling center is represented analytically. The complex dynamic stress intensity factors are evaluated directly from either the stresses or the crack opening displacements of the singular stress term. Numerical examples of cracked anisotropic bimaterial plates are presented to verify the accuracy of the present technique and to provide additions to the very limited number of reference solutions in the literature.     

Transient thermal shock fracture analysis of functionally graded piezoelectric materials by the extended finite element method

Transient thermal dynamic analysis of stationary cracks in functionally graded piezoelectric materials (FGPMs) based on the extended finite element method (X-FEM) is presented. Both heating and cooling shocks are considered. The material properties are supposed to vary exponentially along specific direction while the crack-faces are assumed to be adiabatic and electrically impermeable. A dynamic X-FEM model is developed in which both Crank–Nicolson and Newmark time integration methods are used for calculating transient responses of thermal and electromechanical fields respectively. The generalized dynamic intensity factors for the thermal stresses and electrical displacements are extracted by using the interaction integral. The accuracy of the developed approach is verified numerically by comparing the calculated results with reference solutions. Numerical examples with mixed-mode crack problems are analyzed. The effects of the crack-length, poling direction, material gradation, etc. on the dynamic intensity factors are investigated. It shows that the transient dynamic crack behaviors under the cooling shock differ from those under the heating shock. The influence of the thermal shock loading on the dynamic intensity factors is significant.     

On the complete determination of dynamic states of stress

An attempt is made in dynamic photoelasticity to get the principal directions and principal stresses of a time-dependent two-dimensional state by three simultaneous photoelectric signals. The signals are obtained from birefringence and interference of light in the model itself. The described procedure is possible because a laser was used instead of a conventional light source. It applies essentially to the general case where the principal directions vary as a function of time during a transient state of stress and, probably, it may also be used for certain three-dimensional tests. A comparison with theoretical values in an example has proved a rather good accuracy.     

Exact transient solutions of buried dynamic point forces for elastic bimaterials

The exact transient closed form solutions for applying time dependent point forces at a depth below the interface of dissimilar solids are obtained in this study. The problem is solved by application of Laplace transform methods. The inverse transforms are evaluated by means of Cagniard's method. The equations of wave fronts for reflected and refracted waves induced from the bimaterial interface by the incident waves are determined in this study. The corresponding static solutions of stresses and displacements are reduced from the obtained transient solutions. Numerical results of stress fields during the transient process are obtained for different cases and compared with the corresponding static values.