Acoustoelasticity: Ultrasonic Stress Field Reconstruction by H.R. Dorfi, H.R. Busby and M. Janssen ABSTRACT--Based on the theory of acoustoelasticity, a new ultrasonic stress reconstruction method--the general- ized acoustic ratio (GAR) techniquc ~s developed for lo- cally plane structures and orthotropic materials. For given transit times of the three wave modes and the shear wave polarization angle, the local plane stress tensor is uniquely determined. The GAR technique yields accurate stress es- timates with relatively small temperature sensitivity. Based on calibration constants from three uniaxial specimens, the entire stress field in a compact tension specimen is recon- structed. The results are in very good agreement with stress predictions from an elastoplastic finite element analy- sis. To further improve the measurements, a numerical technique, the stress field approximation (SFA) technique, is developed. The SFA technique uses a smooth local bicu- bic spline approximation and aims at improving the overall stress field estimate by enforcing the equilibrium equations, the stress boundary conditions and symmetry conditions. Numerical results show that both the average error and its spread are indeed reduced. Introduction Since the late 1950s, engineers have used ultrasonic waves to identify stresses in solids. In analogy to photoe- lasticity, the phenomenon was termed the acoustoelastic ef- fect. Compared to diffraction methods, which are commonly used for nondestructive stress measurements, ul- trasonic techniques are both less expensive and faster. However, progress has been slow because of several rea- sons. The relative change in wave speed caused by the stress field is very small, typically of the order of 1 percent or less at the yield point. Furthermore, most engineering materials exhibit slight anisotropy, which causes changes in wave speed comparable to the stress-induced variation. The theory of acoustoelasticity originated from the in- terest in the measurement of third-order elastic constants (TOEC) in crystals. These elastic constants are needed to account for the small nonlinear elastic effects in hypere- lastic solids. Hughes and Kelly~ developed the theory of acoustoelasticity and used ultrasonic waves to determine these constants in crystals by varying the applied stress. Toupin and Bernstein 2 extended their work to hyperelastic materials with arbitrary symmetry. Benson 3 and Bergman 4 H.R. Dorfi is a Graduate Student, and H.R. Busby is Professor, Depart- ment of Mechanical Engineering, The Ohio State University, Columbus, OH 43210. M. Janssen is Professor, Department of Materials Science, Delft University of Technology, Delft, The Netherlands. Original manuscript submitted." July 20, 1994. Final manuscript received: August 1, 1995. were the first engineers to employ ultrasonic waves for stress measurements in solids. Most early experimenters assumed isotropic material properties; however, from experiments it was found that in- itial weak anisotropy and slight inhomogeneity in the ma- terial cause large errors in the stress prediction. To address this problem, Tokuoka5'6 and Iwashimizu7'8 developed a the- ory for slightly inhomogeneous predeformation and anisot- ropic materials. Since texture of materials is mainly caused by processes such as rolling or drawing, material symme- tries do in general exist and can be adequately described as orthotropic. Theories for orthotropic materials were de- veloped by King and Fortunko 9 and Pao and Gamer. 1~ It should be noted that orthotropic materials have 9 second- order elastic constants and 20 third-order elastic constants. Thus, in general, one wants to avoid measurement of all these constants. As will be shown later, acoustic birefrin- gence only requires the measurement of three material specific acoustoelastic constants and one initial texture birefrin- gence parameter, even for strongly orthotropic materials (compared to one parameter for strictly isotropic materials). Most acoustoelastic theories are based on the assumption that the body is in a hyperelastic state of stress; while this assumption is satisfied frequently, it is not valid for stresses induced by local plastic deformation because of the local- ized change in material parameters. Therefore, the identi- fication of residual stresses caused by plastic deformation has been less reliable. 11Furthermore, most materials exhibit slight anisotropy or texture due to the fabrication process. The change in wave speed is affected not only by stresses but also by this anisotropy. Since both stress- and texture- induced velocity shifts are usually of similar magnitude, it has been difficult to separate them. Recently, Man and Lu ~2 proposed an acoustoelastic the- ory within the framework of linear elasticity with initial stress. In contrast to earlier research, the authors do not use any reference to an initial, undeformed state. Thus their theory is not limited to materials, which are predeformed hyperelastically. The theory only requires that the superim- posed ultrasonic wave causes elastic deformation only. This certainly holds true for low-amplitude ultrasonics. Their ap- proach is thus valid for both elastic and plastic predefor- mation. Based on a work by Hoger 13 and Biot's work on prestressed bodies, 14 a general equation is derived which depends only on the wave speeds and the density of the material. The advantage of this formulation is that no as- sumptions are made about the origin of the residual stresses; they are treated as part of the constitutive equa- tion. Their equation is limited to orthotropic solids with co- inciding material principal stress axes. A minimum of four Experimental Mechanics 325