Transition from Flat to Slant Fracture in Ductile Materials

Abstract

We investigate ductile fracture in an aluminum alloy which exhibits an interesting switch in mode from flat fracture to slant fracture. While this transition is typically considered to be triggered by a change in crack tip constraint with specimen thickness, we explore this transition in rolled sheet material simply by controlling the strain hardening behavior of the material. Specifically, experiments are performed on two different heat-treatments of the same alloy, resulting in two materials that differ only in their strain-hardening behavior. Based on an examination of the microscopic and macroscopic strain evolution, we conclude that the transition arises because of strain localization that precedes fracture.

Appendix: Determination of the Stress from Strain Measurements

The conversion of the strain measurements to stress requires a material model; the J₂ deformation theory of plasticity is taken to be the appropriate model. The effective stress is defined as $\sigma_{e} = \sqrt {3J_{2} } = \left( {3s_{ij} s_{ij} /2} \right)^{1/2}$ where $s_{ij}$ are the components of the stress deviator, and the plastic equivalent strain is defined as $\bar{\varepsilon } = \left( {2\varepsilon_{ij} \varepsilon_{ij} /3} \right)^{1/2}$. The variation of the effective stress with the plastic equivalent strain is obtained from a uniaxial test and is indicated by $\sigma \left( {\bar{\varepsilon }} \right)$. The von Mises yield criterion is then expressed as

$$\Phi \left( {J_{2} ,\bar{\varepsilon }} \right) = 3J_{2} - \sigma^{2} \left( {\bar{\varepsilon }} \right) = 0$$

(2)

The total strain is decomposed additively into the plastic and elastic components; the plastic strain develops in the direction of the normal to the yield surface and can then be expressed as:

$$\varepsilon_{ij}^{p} = d\lambda s_{ij}$$

(3)

The stress-strain relationship is written as:

$$\varepsilon_{ij} = \frac{1 + v}{E}\sigma_{ij} - \frac{v}{E}\sigma_{kk} \delta_{ij} + \frac{3}{2}\left( {\frac{1}{{E_{s} }} - \frac{1}{E}} \right)s_{ij}$$

(4)

where E _s is the secant modulus. The first two terms make up the linear elastic component of strain, and the third term is the plastic component. For conditions of plane stress imposing $\sigma_{33} = 0$, Eq. (4) can be inverted and written explicitly as

$$\begin{aligned} \sigma_{11} & = \frac{{E_{s} }}{{1 - \nu_{s}^{2} }}\left[ {\varepsilon_{11} + \nu_{s} \varepsilon_{22} } \right] \\ \sigma_{22} & = \frac{{E_{s} }}{{1 - \nu_{s}^{2} }}\left[ {\varepsilon_{22} + \nu_{s} \varepsilon_{11} } \right] \\ \sigma_{12} & = \frac{{E_{s} }}{{1 + \nu_{s}^{{}} }}\varepsilon_{12} \\ \varepsilon_{33} & = - \frac{{\nu_{s} }}{{1 - \nu_{s} }}\left[ {\varepsilon_{11} + \varepsilon_{22} } \right] \\ \end{aligned}$$

(5)

where

$$\nu_{s} = \frac{1}{2} + \frac{{E_{s} }}{E}\left( {\nu - \frac{1}{2}} \right)$$

(6)

In order to determine the stress components, it is necessary to determine E _s and $\nu_{s}$ with increasing strain. In the experiments, we determine the displacement vector ${\mathbf{u}}({\mathbf{x}})$ at every point in the field of view using digital image correlation. Using the strain-displacement relations, we can determine the strain field ${\varvec{\upvarepsilon}}({\mathbf{x}})$. Next, we need to determine the stress field for a nonlinear material; this is accomplished by first estimating the effective stress. First, we calculate ${\varvec{\upvarepsilon}}:{\varvec{\upvarepsilon}}$ from Eq. (4) and simplify to get

$$\frac{2}{3}\left[ {\frac{1 + \nu }{E} + \frac{3}{2}\left( {\frac{{E - E_{s} }}{{E_{s} E}}} \right)} \right]^{2} \sigma_{e}^{2} = \left( {\varepsilon_{ij} \varepsilon_{ij} - \frac{1}{3}\varepsilon_{kk}^{2} } \right)$$

(7)

Since E _s in the left hand side is a function of $\sigma_{e}$, the above represents a nonlinear equation that can be solved for the effective stress $\sigma_{e} \left( {\mathbf{x}} \right)$ from the measured strain field. This allows for conversion of the experimental measurements into contours of the Mises stress field. In particular, this can be used in a crack problem to determine the plastic zone boundary. Then E _s and $\nu_{s}$ can be determined for the given material properties. Subsequently, the stress field ${\varvec{\upsigma}}({\mathbf{x}})$ can be determined through Eq. (5). We illustrate this procedure here for the Ramberg-Osgood material model in Eq. (1). The secant modulus is a function of the stress level; this is found easily from Eq. (1):

$$E_{s} \left( {\sigma_{e} } \right) = E\left[ {1 + \alpha \left( {\frac{{\sigma_{e} }}{{\sigma_{Y} }}} \right)^{n - 1} } \right]^{ - 1}$$

(8)

One can work out a similar expression for other models of uniaxial constitutive response. Substituting for strains in the principal orientation, and utilizing the Ramberg-Osgood material model, Eq. (7) can be expanded to the following equation for the effective stress:

$$\begin{array}{*{20}l} {\frac{2}{3}\left[ {1 + \nu + \frac{3}{2}\alpha \left( {\frac{{\sigma_{e} }}{{\sigma_{y} }}} \right)^{n - 1} } \right]^{2} \frac{{\sigma_{e}^{2} }}{{E^{2} }} - \left( {\varepsilon_{1}^{2} + \varepsilon_{2}^{2} } \right)} \hfill \\ \; { - \left( {\varepsilon_{1} + \varepsilon_{2} } \right)^{2} \frac{{\left\{ {\alpha \left( {\frac{{\sigma_{e} }}{{\sigma_{y} }}} \right)^{n - 1} + \, 2\nu } \right\}^{2} - \frac{4}{3}\left( {1 \, - \, 2\nu } \right)^{2} }}{{\left\{ {\alpha \left( {\frac{{\sigma_{e} }}{{\sigma_{y} }}} \right)^{n - 1} + \, 2\left( {1 - \nu } \right)} \right\}^{2} }} = 0} \hfill \\ \end{array}$$

(9)

Equation (9) can be solved numerically for the equivalent stress at each point in the field with the measured values of the strains at each time step. This value of $\sigma_{e}$ can be used in Eqs. (8) and (6) to calculate E _s and $\nu_{s}$, respectively, and then used in Eq. (5) to determine all components of stress.

Once the strain field and the stress field on the surface are determined, the J-integral can be evaluated numerically

$$J = \int\limits_{\Gamma } {\left( {Wdy - T_{\alpha } \frac{{\partial u_{\alpha } }}{{\partial x_{1} }}ds} \right)} = \sum {W\varDelta y - \sum {T_{\alpha } \frac{{\partial u_{\alpha } }}{{\partial x_{1} }}\varDelta s} }$$

(10)

W is the strain energy, T _α are the components of the traction vector, u _α are the displacement components and ds is the length increment along the contour Γ. In addition to evaluating the J-integral, multiple contours around the crack tip can be defined, and the path independence of the J-integral can be examined.