Piercing soft solids: A mechanical theory for needle insertion
Introduction
Biomedical operations such as surgery, drug injection (Abolhassani et al., 2006), biopsy (Giovannini et al., 2017), and blood sampling are achieving growing importance in modern medicine especially when automated (Wei et al., 2004). However, their success and safety depend on our ability to predict the behavior of biological tissue when rupturing under deep indentation. This is also crucial for understanding the biomechanics of soft tissue injury. In addition to medical applications, a precise knowledge of the mechanisms underlying the phenomenon of cutting and piercing is important for material characterization (Azar and Hayward, 2008; Barney et al., 2019), ballistic protection (Nyanor et al., 2018), manufacturing (Long et al., 2014), and food processing (Kamyab et al., 1998).
Throughout evolution, all animal species have evolved with the ability to pierce through and break down biological tissue to feed and defend. The morphology of beaks (Fritz et al., 2014), claws (Lautenschlager, 2014), nails (Tsang et al., 2019), quills (Cho et al., 2012), and teeth (Conith et al., 2016) has ultimately reached remarkable geometrical and mechanical properties to ensure success in cutting or piercing with the intended precision. Cutting and piercing, however, have been mainly explored empirically and a comprehensive theory to mathematically describe these processes is currently missing. This is due to the complexity of the mechanical problem, which involves large deformations, large strains, and a complex fracture mechanism. Current models for nonlinear elastic fracture mechanics provided solutions for traditional uniaxial and pure-shear stress states (Long and Hui, 2015). However, a material being cut is subject to a more complex stress state, not described in previous fracture mechanics models.
In this paper, we focus on the mechanical problem of piercing of a soft solid body with a needle, i.e. puncture. The first experimental investigations on the mechanics of puncture were performed on rubber (Stevenson and Ab-Malek, 1994), followed by the first theoretical investigations (Shergold and Fleck, 2004). The latter involved the calculation of the critical force required to deeply penetrate a soft material with needles having a flat or conical tip after the needle is already inserted. This force depends on the toughness of the material, its stiffness, and the radius of the needle. The same authors validated their theory by piercing rubber and porcine skin with needles having various geometries and sizes (Shergold and Fleck, 2005). However, they did not calculate the critical force required for needle insertion, before deep penetration begins. Previous experiments on rubber (Stevenson and Ab-Malek, 1994; Shergold and Fleck, 2005; Yeh and Livingston, 1961; Ab-Malek and Stevenson, 1984), biological tissue (Okamura et al., 2004), and silicone gel (Das and Ghatak, 2011) evidenced a force peak at needle insertion, followed by a force drop once the needle had pierced through the surface of the specimen and deeply penetrated it.
To unravel the determinants of needle insertion, (Fakhuri et al., 2015) performed an experimental investigation using needles of various geometries and sizes, puncturing gels with various stiffness and toughness. They described the relation between puncture force and needle radius with two regimes. For small needle radii, they observed the energy-limited correlation , with the toughness of the material. However, measured from puncture experiments is significantly larger than that measured with traditional fracture tests. For larger needle radii, they observed the stress-limited correlation , with a ‘cohesive stress’. This mechanical property, however, is not measured with other experiments for comparison. Hence, a physical model capable of predicting the condition of needle insertion from material parameters and needle geometry is currently missing.
To overcome the abovementioned limitations, we propose a simple mechanical theory based on a minimum energy principle considering two needle-specimen configurations: indentation and penetration. Before needle insertion, the needle simply indents the material and the energetic cost associated with it only depends on elastic deformation. At needle insertion, the needle pierces through the material and deeply penetrates it. Needle insertion occurs when penetration suddenly becomes the energetically favored mechanism over indentation. Our theory is detailed in the next section. We will then provide some numerical results and compare our model against experiments.
Our theory is based on an energetic comparison between two distinct needle-specimen configurations, namely (i) indentation and (p) penetration. Fig. 1a provides a sketch of configuration (i), while Fig. 1b-c provides a sketch of (p). The transition between the two configurations occurs at needle insertion. The needle has a cylindrical stalk with cross-sectional radius and a spherical tip having the same radius. The axis of the needle is orthogonal to the free surface of the specimen; is the displacement of the needle tip toward the specimen and is the force applied to the back of the needle, aligned with the displacement . We consider the specimen to be significantly larger than , so that the specimen size can be ignored. The force applied to the back of the needle is given by the function with . The mechanical work done by the force to push the needle to the depth is
The forces and are applied to the needle in the configurations (i) and (p), respectively. By replacing these forces in Eq. (1) we obtain the mechanical work associated with the two configurations as and . The evolution of , , , and for an incrementing depth is sketched in Fig. 2. As evidenced in this figure, needle insertion occurs when reaches the critical depth , at which . For we have , hence (i) is the energetically favoured configuration (Fig. 2b). Conversely, for we have , hence (p) is the energetically favoured configuration (Fig. 2b). The critical force for needle insertion is the force in configuration (i) at , hence as evidenced in Fig. 2a. This force corresponds to the slope of the curve versus in Fig. 2b at . Once the needle is inserted, the penetration force required to push the needle further down into the specimen is and is equivalent to the slope of the curve versus in Fig. 2b. is constant since we neglect friction and adhesion between the specimen and the needle.
We describe the mechanical behavior of the material with an incompressible single-term Ogden strain energy density functional (Ogden, 1972)
In this equation, is the shear modulus of the material (with Young modulus ); is a dimensionless material parameter indicating the tendency of the material to strain-harden; and and are the principal stretches. The symmetry of Eq. (2) with respect to the three principal directions 1, 2 and 3 is associated with the isotropic behavior of the material. For , Eq. (2) gives the neoHookean form. For larger the model adopted becomes more representative of rubbers and biological materials, which exhibit a typical J-shaped force-displacement curve under uniaxial tensile test. Most biological materials however can exhibit anisotropic behavior; hence Eq. (2) might become unsuitable when a strongly anisotropic behavior is observed. For all other cases, Eq. (2) provides a good generalization of cases by choosing the proper value of and . The Cauchy (true) stress in the material in direction 1 iswith the hydrostatic pressure applied to the material, i.e. a Lagrange multiplier enforcing incompressibility. Eq. (3) can be rewritten for directions 2 and 3 in the same way thanks to the isotropic behavior of the material.
The needle-specimen system can be described by the dimensionless displacement and the dimensionless force . Their relation cannot be obtained analytically for either (i) and (p) configurations. We, therefore, adopt a numerical approach based on finite element analysis (FEA).
Indentation configuration (i)
The relationship between and is calculated via FEA, as detailed in this section. This relation can be generally represented with a series of power-law terms aswith and the power-law-series coefficients, functions of the variable . As reported in Appendix A, the results from FEA can be fitted with Eq. (4) with just one power law-term. In this case, we have and , which values are reported in Table 1 for various .
For very small indentation depths, and the Hertzian relationship for linear elastic spherical indentation should apply, giving (Johnson, 1985). This would require Eq. (4) to match this solution at . However, when the surface of the spherical tip is not entirely touching the material while experiments showed that needle insertion occurs at (Fakhuri et al., 2015), when the entire surface of the tip is in contact with the material. The same authors also observed that the influence of tip geometry on the material response to indentation reduces significantly at large indentation depths (i.e. at ), hence giving more generality to our theory. By substituting the force at Eq. (4) into Eq. (1), we obtain the dimensionless mechanical work
The coefficients and in Eq. (4) are obtained from FEA under quasi-static conditions using the commercial software ANSYS. Given the radial symmetry of the problem and the incompressibility of the material, we used the radially symmetric planar elements 182 based on hybrid formulation (see ANSYS’ manual for more details). Fig. 3a sketches the boundary conditions used in the model, with external boundaries fixed. The cylindrical specimen has radius and height . To neglect the influence of the specimen size in our results, we tested various values of and . For a maximum depth , we concluded that a specimen size of was sufficient to remove its dependency on the results with 1% accuracy. Fig. 3b shows the finite element mesh adopted. We observed a significant gradient in the strain energy density of the material in the proximity of the contact region between needle and specimen, as shown in Fig. 3c. Due to this observation, we adopted a finer mesh near the contact region and a coarser one in the remote regions, near the fixed boundary. The contact between the indenter and the specimen is considered frictionless.
Penetration configuration (p)
The mechanical work required for the needle to penetrate through the material is composed of two main contributions, the work of fracture and the work of spacing . The former is the energetic cost required to nucleate and propagate a crack underneath the indenter while the latter is the energetic cost required to ‘space out’ the material so that the needle can slide into the crack. Both energetic contributions depend on the mechanism of needle penetration. (Shergold and Fleck, 2004; Shergold and Fleck, 2005) observed two failure mechanisms, based on the shape of the indenter. Conical indenters create a planar crack, which propagates in Mode I and is parallel to the axis of the needle. Flat punch indenters instead rupture the material underneath by creating a ring-shaped crack, which propagates in Mode II. This occurs thanks to a shear stress concentration at the perimeter of the contact region, typical of flat indenters. In some cases, however, (Lin et al., 2009) observed that flat indenters can penetrate the specimen through a planar crack as described for conical indenters. This is because Mode II fracture toughness is commonly much larger than in Mode I.
No specific study unraveled the penetration mechanism of spherical indenters. However, from the observations above, we assume that spherical indenters penetrate the specimen through a planar crack in Mode I, as described in Fig. 1b-c.
The work of fracture is given bywhere is the area of the crack in Fig. 1b-c, and is the toughness of the material. The ratio depends from the properties of the material and scales with the ratio . For very soft materials, experimental observations reported large indentation depths at puncture, , compared to the radius of the needle (Fakhuri et al., 2015). This observation validates the hypothesis of , allowing us to neglect the second term in the parenthesis in Eq. (6).
The work of spacing, considering the penetration mechanism described in Fig. 1b-c, is given by the mechanical work required to open the crack so that the needle can slide into it. This can be written aswith a dimensionless parameter, function of the ratio . The relation between and is calculated via FEA by (Shergold and Fleck, 2004). The total work required for the needle to penetrate through the specimen at the depth is finally calculated by summing the contributions from Eq. (6), with , and (7). This gives, in its dimensionless form,
By differentiating Eq. (8) with respect to we obtain the dimensionless penetration force
In Eq. (8) and (9), the dimensionless crack size is unknown and should be determined in relation to the material parameters. For a given choice of and , the right-hand side of Eq. (9) presents a global minimum in the variable , at . Following the principle of minimum energy, we assume the minimum force at to be that at which the needle penetrates the specimen. I.e. a small crack can propagate unstably and increase its size until this reaches , after which the crack stops. The relation between and , for various values of , can only be calculated numerically via FEA and has been obtained by (Shergold and Fleck, 2004). To obtain an explicit relation, we fitted the numerical results to the functionwith the nominal crack size, obtained when , and and material coefficients that depend on , as reported in Table 2.
Take now . The relation between and is also extrapolated numerically from (Shergold and Fleck, 2004), and fitted to the functionwith , , , , and material coefficients given in Table 2 as a function of . All the parameters reported in Table 2 are obtained with the method of the least squares, giving a maximum error comprised within 1%.
By substituting and into Eq. (8) we obtain the mechanical work required for needle penetration .
Equating the result with , with taken from Eq. (5), we obtain the critical depth aswith and taken from Eq. (11) and (12). Substituting this into Eq. (4) we obtain the critical force for needle insertion as
The dimensionless force required to penetrate the material after needle insertion can be calculated from Eq. (9) by replacing and with and , respectively. Comparing this force with Eq. (13), we can conclude that the dimensionless force drop, , produced by needle insertion obeys the simple relation .
Section snippets
Results
The critical conditions for needle insertion are described by the dimensionless critical depth , given by Eq. (12), and the dimensionless critical force , given by Eq. (13). These are functions of the dimensionless parameter , via Eq. (10) and (11), and the material parameter (Table 1, Table 2). Fig. 4 reports versus , for various , in a log-log plot, while Fig. 5 reports in the same way. As shown in these figures, and are proportional to the
Discussion and conclusions
In the proposed theory, the complex mechanism of needle insertion is simply described by a sharp transition between needle indentation and needle penetration, a process driven by elastic instability. Our theory is based on perfect energy transfer, i.e. it relies on the hypothesis that all the strain energy accumulated in the material during indentation, , is immediately and entirely available to nucleate and propagate the crack that serves as a channel to accommodate the penetration of the
Declaration of Competing Interest
None.
Acknowledgments
We thank David Labonte (Imperial College, London) for helpful conversations. This work was supported by the Department of National Defense (DND) of Canada (CFPMN1–026), by the Natural Sciences and Engineering Research Council of Canada (NSERC) (RGPIN-2017–04464), and by the Human Frontiers in Science Program (RGY0073/2020).
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