Noninvasive measurement of local stress inside soft materials with programmed shear waves.

Mechanical stresses across different length scales play a fundamental role in understanding biological systems' functions and engineering soft machines and devices. However, it is challenging to noninvasively probe local mechanical stresses in situ, particularly when the mechanical properties are unknown. We propose an acoustoelastic imaging-based method to infer the local stresses in soft materials by measuring the speeds of shear waves induced by custom-programmed acoustic radiation force. Using an ultrasound transducer to excite and track the shear waves remotely, we demonstrate the application of the method by imaging uniaxial and bending stresses in an isotropic hydrogel and the passive uniaxial stress in a skeletal muscle. These measurements were all done without the knowledge of the constitutive parameters of the materials. The experiments indicate that our method will find broad applications, ranging from health monitoring of soft structures and machines to diagnosing diseases that alter stresses in soft tissues.

Non-destructive mapping of stress and strain in soft thin films through sound waves.

Measuring the in-plane mechanical stress in a taut membrane is challenging, especially if its material parameters are unknown or altered by the stress. Yet being able to measure the stress is of fundamental interest to basic research and practical applications that use soft membranes, from engineering to tissues. Here we present a robust non-destructive technique to measure directly in-situ stress and strain in soft thin films without the need to calibrate material parameters. Our method relies on measuring the speed of elastic waves propagating in the film. Using optical coherence tomography, we verify our method experimentally for a stretched rubber membrane, a piece of cling film (about 10 µm thick), and the leather skin of a traditional Irish frame drum. We find that our stress predictions are highly accurate and anticipate that our technique could be useful in applications ranging from soft matter devices to biomaterial engineering and medical diagnosis.

An ultrasonic method to measure stress without calibration: The angled shear wave method.

Measuring stress levels in loaded structures is crucial to assess and monitor structure health and to predict the length of remaining structural life. Many ultrasonic methods are able to accurately predict in-plane stresses inside a controlled laboratory environment but struggle to be robust outside, in a real-world setting. That is because these methods rely either on knowing beforehand the material constants (which are difficult to acquire) or require significant calibration for each specimen. This paper presents an ultrasonic method to evaluate the in-plane stress in situ directly, without knowing any material constants. The method is simple in principle, as it only requires measuring the speed of two angled shear waves. It is based on a formula that is exact for incompressible solids, such as soft gels or tissues, and is approximately true for compressible "hard" solids, such as steel and other metals. The formula is validated by finite element simulations, showing that it displays excellent accuracy, with a small error on the order of 1%.

A proof that multiple waves propagate in ensemble-averaged particulate materials.

Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterize wave propagation through an inhomogeneous material, the most crucial parameter is the effective wavenumber. For this reason, there are many published studies on how to calculate a single effective wavenumber. Here, we present a proof that there does not exist a unique effective wavenumber; instead, there are an infinite number of such (complex) wavenumbers. We show that in most parameter regimes only a small number of these effective wavenumbers make a significant contribution to the wave field. However, to accurately calculate the reflection and transmission coefficients, a large number of the (highly attenuating) effective waves is required. For clarity, we present results for scalar (acoustic) waves for a two-dimensional material filled (over a half-space) with randomly distributed circular cylindrical inclusions. We calculate the effective medium by ensemble averaging over all possible inhomogeneities. The proof is based on the application of the Wiener-Hopf technique and makes no assumption on the wavelength, particle boundary conditions/size or volume fraction. This technique provides a simple formula for the reflection coefficient, which can be explicitly evaluated for monopole scatterers. We compare results with an alternative numerical matching method.

Reflection from a multi-species material and its transmitted effective wavenumber.

We formally deduce closed-form expressions for the transmitted effective wavenumber of a material comprising multiple types of inclusions or particles (multi-species), dispersed in a uniform background medium. The expressions, derived here for the first time, are valid for moderate volume fractions and without restriction on the frequency. We show that the multi-species effective wavenumber is not a straightforward extension of expressions for a single species. Comparisons are drawn with state-of-the-art models in acoustics by presenting numerical results for a concrete and a water-oil emulsion in two dimensions. The limit of when one species is much smaller than the other is also discussed and we determine the background medium felt by the larger species in this limit. Surprisingly, we show that the answer is not the intuitive result predicted by self-consistent multiple scattering theories. The derivation presented here applies to the scalar wave equation with cylindrical or spherical inclusions, with any distribution of sizes, densities and wave speeds. The reflection coefficient associated with a halfspace of multi-species cylindrical inclusions is also formally derived.