[Paper Review] Adaptation of the tapered double cantilever beam test for the measurement of fracture energy and its variations with crack speed

Aditya Vasudevan, Thiago Melo Grabois+4 more|arXiv (Cornell University)|Jan 12, 2021

Force Microscopy Techniques and Applications12 references4 citations

TL;DR

This paper proposes a modified Tapered Double Cantilever Beam (TDCB) test geometry that enables stable crack growth in brittle materials like PMMA, allowing precise measurement of fracture energy and its rate-dependent variations. By exploiting an exponential compliance increase with crack length, the method enables accurate determination of the kinetic law $G_c(v)$ through analytical modeling and global optimization, achieving high-fidelity predictions of crack speed, energy release rate, and displacement response without reliance on finite element simulations for parameter extraction.

ABSTRACT

In this work we present the design of a new test geometry inspired by the Tapered Double Cantilever Beam (TDCB) specimen that is shown to provide an improved characterization of the fracture properties of brittle solids. First, we show that our new design results in an exponential increase of the specimen compliance with crack length, leading to an extremely stable crack growth during the test. We determine an analytical description of this behavior, which provides a simple procedure to extract the fracture energy without depending on finite element calculations. Validation tests are done on polymethylmethacrylate (PMMA) specimens. We use both finite element simulations and our analytical model to interpret the data. We find a very good agreement between the toughness determined by both methods. The stable nature of crack growth in our improved TDCB specimens results in a precise control of the crack speed. This feature is employed to go one step further and characterize the variations of toughness with crack speed. We propose an original optimization procedure for the determination of the material parameters characterizing the kinetic law describing the toughness rate dependency. Overall, the approach proposed together with the newly designed test geometry offer unprecedented possibilities for the full and accurate characterization of the fracture behavior of brittle materials such as rocks, sandstone, mortar etc.

Motivation & Objective

• To address the challenge of unstable crack propagation in standard fracture tests for brittle solids, which hinders accurate toughness measurement.
• To develop a modified TDCB geometry that ensures stable, controlled crack growth over a wide range of crack speeds.
• To enable accurate measurement of fracture energy $G_c$ and its variation with crack speed $v$ without relying on finite element simulations for parameter extraction.
• To establish a global optimization procedure for determining the kinetic law $G_c(v)$ that describes rate-dependent fracture toughness.
• To validate the method experimentally on PMMA and demonstrate its predictive capability for crack evolution and energy release rate.

Proposed method

• Design of a modified TDCB specimen with a width-to-length ratio near unity, eliminating the straight portion and leading to an exponential increase in compliance with crack length.
• Derivation of an analytical model for the compliance and energy release rate, showing exponential decay of $G$ with crack length under fixed displacement, ensuring crack stability.
• Use of digital image correlation (DIC) to measure crack length and displacement during testing, enabling precise tracking of crack tip position and crack speed.
• Implementation of a global optimization procedure to fit the experimental force-displacement curves to theoretical predictions based on linear elastic fracture mechanics and the kinetic law $G_c(v) = G_c^0 \exp(\gamma v)$.
• Validation of the analytical model and optimization results using both finite element simulations and experimental data on PMMA specimens.
• Determination of material parameters $G_c^0$ and $\gamma$ by minimizing the error between predicted and experimental force-displacement responses across multiple loading rates.

Experimental results

Research questions

• RQ1 Can a modified TDCB geometry be designed to achieve stable crack growth in brittle materials like PMMA?
• RQ2 Does the modified geometry lead to an exponential compliance increase with crack length, enabling stable energy release rate decay?
• RQ3 Can the fracture energy $G_c$ be accurately extracted without finite element analysis using the derived analytical model?
• RQ4 Can the kinetic law $G_c(v) = G_c^0 \exp(\gamma v)$ be reliably determined from experimental data using a global optimization procedure?
• RQ5 To what extent can the model predict crack evolution, velocity, and energy release rate across different crack speeds?

Key findings

• The modified TDCB geometry induces an exponential increase in compliance with crack length, resulting in a strong stabilization of crack propagation and enabling precise control of crack speed.
• The analytical model accurately predicts the force-displacement response, with excellent agreement between experimental data and simulations, validating the theoretical framework.
• The optimized parameters for PMMA were found to be $G_c^0 = 2000$ J/m² and $\gamma = 0.20$, with uncertainties of $\pm 100$ J/m² and $\pm 0.01$, respectively.
• The kinetic law $G_c(v) = G_c^0 \exp(\gamma v)$ was successfully determined from a single experiment at $\dot{\delta} = 2.5$ μm/s, yielding $G_c^0 = 1750$ J/m² and $\gamma = 0.192$, demonstrating the feasibility of single-test parameter identification.
• Predicted crack length $c(t)$, crack speed $v(c)$, and energy release rate $G(c)$ from the kinetic law showed excellent agreement with experimental measurements, confirming the model's predictive power.
• The method enables accurate characterization of fracture energy and its rate dependence across a wide range of crack speeds, offering a robust alternative to conventional fracture tests for brittle materials.