16.2.3 Jardine Model

The model proposed by Jardine et al. [50] is based on the relation between the secant Young's modulus and the axial strain, measured in an undrained triaxial compression test, i.e., a test in which a cylindrical specimen is loaded by an increasing axial load while the radial stress is kept constant, see for instance Jardine et al. [51]. The secant Young's modulus is defined as:

$${\frac{{ \sigma _{a} - \sigma _{a;0} }}{{ \varepsilon _{a} }}}$$ (16.13)

In which E_u is the secant Young's modulus, $\varepsilon_{{a}}^{}$ is the axial strain, $\sigma_{{a}}^{}$ is the axial stress, and $\sigma_{{a;0}}^{}$ is the initial axial stress (for which $\varepsilon_{{a}}^{}$ = 0 ). In the triaxial test axial stresses are measured. With (16.13) E_u follows directly from these measurements.

The basic assumption of the Jardine model is that the relation between the secant Young's modulus and the axial strain over the elastic range can be expressed in the form:

$$\biggl( \alpha \left(\vphantom{ \log \frac{ \varepsilon _{a} }{ C } }\right. {\frac{{ \varepsilon _{a} }}{{ C }}} \left.\vphantom{ \log \frac{ \varepsilon _{a} }{ C } }\right)^{{\!\gamma}}_{} \biggr)$$ (16.14)

This relation can be visualized as a stretched periodic function in a diagram of E_u vs. $\varepsilon_{{a}}^{}$ [Fig.16.2].

Figure 16.2: Jardine parameters in stiffness-log(strain) diagram

The relation contains five independent parameters (C , D , E , F , and G ) which are considered to be material constants. These parameters can be read directly from the stiffness-strain diagram. Parameters D and E are intermediate and can be applied to derive $\alpha$ and $\gamma$ using the following formulae, assuming that the angular part in (16.14) has to be equal to ${\tfrac{{1}}{{2}}}$$\pi$ at the medium stiffness and equal to $\pi$ at the minimum stiffness:

$$\gamma {\frac{{ \log 2 }}{{ \log\!\left( \dfrac{ \log( E / C )}{ \log( D / C) } \right) }}} \alpha {\frac{{ \tfrac{1}{2}\pi }}{{ \Bigl( \log( D / C ) \Bigr)^{\!\gamma} }}}$$ (16.15)

F and G are not necessarily equal to the maximum and medium measured stiffness. F is the maximum of the curve that fits the test data best; this may be a projected maximum outside the range of data. The projected minimum stiffness may even have a negative value, as long as E_u is positive in the specified range over which (16.14) is valid. This range is bounded by $\varepsilon_{{\mathrm{min}}}^{}$ and $\varepsilon_{{\mathrm{max}}}^{}$ . Beyond these boundaries the tangent Young's modulus is assumed to be constant. For strain below $\varepsilon_{{\mathrm{min}}}^{}$ this implies a constant secant Young's modulus, which is not the case for strain beyond $\varepsilon_{{\mathrm{max}}}^{}$ . A practical value for $\varepsilon_{{\mathrm{min}}}^{}$ is the smallest strain for which test data is available. For $\varepsilon_{{\mathrm{max}}}^{}$ care is required to ensure compatibility with the onset of plastic yield. With a high value for $\varepsilon_{{\mathrm{max}}}^{}$ , a negative elastic tangential stiffness may occur, causing numerical instability, even when $\varepsilon_{{\mathrm{max}}}^{}$ is chosen less than E .

In the DIANA implementation the Jardine model is generalized by substitution of the deviatoric strain invariant:

$$\varepsilon_{{\mathrm{eq}}}^{} \sqrt{{ \tfrac{2}{3}\Bigl( ( \varepsilon _{1} - \varepsilon _{2} ... ... \varepsilon _{3} )^{2} + ( \varepsilon _{3} - \varepsilon _{1} )^{2} \Bigr) }}$$ (16.16)

for $\varepsilon_{{a}}^{}$$\sqrt{{ 3 }}$ , with $\varepsilon_{{1}}^{}$ , $\varepsilon_{{2}}^{}$ , and $\varepsilon_{{3}}^{}$ defined as the principal elastic strains. The origin of the factor $\sqrt{{ 3 }}$ can be traced by substituting the stress state of the undrained triaxial test ( $\varepsilon_{{1}}^{}$ = $\varepsilon_{{a}}^{}$ , $\varepsilon_{{2}}^{}$ = $\varepsilon_{{3}}^{}$ = - ${\tfrac{{1}}{{2}}}$$\varepsilon_{{a}}^{}$ ) into (16.16). From (16.14) an expression for the tangent Young's modulus E_ut can be derived. After substitution of $\varepsilon_{{\mathrm{eq}}}^{}$ these expressions can be written as:

$$\varepsilon_{{\mathrm{eq}}}^{} \alpha \gamma$$ E_u (16.17)

$$\varepsilon_{{\mathrm{eq}}}^{} \alpha \gamma {\frac{{ ( F - G ) \, \alpha \, \gamma \, I^{\gamma - 1} }}{{ 2.303 }}} \alpha \gamma$$ E_ut (16.18)

with

$$\left(\vphantom{ \frac { \varepsilon _{\mathrm{eq}} }{ \sqrt{ 3 } \: C } }\right. {\frac{{ \varepsilon _{\mathrm{eq}} }}{{ \sqrt{ 3 } \: C }}} \left.\vphantom{ \frac { \varepsilon _{\mathrm{eq}} }{ \sqrt{ 3 } \: C } }\right)$$ (16.19)

The equivalent elastic strains corresponding to the boundaries $\varepsilon_{{\mathrm{eq;min}}}^{}$ and $\varepsilon_{{\mathrm{eq;max}}}^{}$ are:

$$\varepsilon_{{\mathrm{eq;min}}}^{} \varepsilon_{{\mathrm{min}}}^{} \sqrt{{ 3 }} \varepsilon_{{\mathrm{eq;max}}}^{} \varepsilon_{{\mathrm{max}}}^{} \sqrt{{ 3 }}$$ (16.20)

With the assumption of a constant tangent Young's modulus outside the boundaries, the general expression for E_u becomes:

$$\begin{cases} f_{1} ( \varepsilon _{\mathrm{eq;min}} ) & \qu... ...m{eq}} > \varepsilon _{\mathrm{eq;max}}$}\\ [1.5ex] \end{cases}$$ (16.21)

For each iteration, in each integration point, (16.21) is used to calculate the updated stresses. To be able to execute analyses with initial stresses unequal to zero, the stresses have to be updated by addition of stress increments. The stress increment is found by calculating the averaged tangent stiffness over $\Delta$$\varepsilon_{{\mathrm{eq}}}^{}$ , which can be done exactly with the given definition of the secant stiffness (16.13). We will explain this first for the triaxial test and then generalized.

The relation between the tangent and secant Young's modulus for the triaxial test is:

$${\frac{{ \,\mathrm{d}\sigma _{a} }}{{ \,\mathrm{d}\varepsilon _{a} }}} {\frac{{ \,\mathrm{d}( E_{\mathrm{u}} \, \varepsilon _{a} ) }}{{ \,\mathrm{d}\varepsilon _{a} }}}$$ (16.22)

With the given relation between axial strain and the secant Young's modulus, the increment in axial stress $\Delta$$\sigma_{{a}}^{}$ , caused by a prescribed increment in axial strain $\Delta$$\varepsilon_{{a}}^{}$ , is:

$$\Delta \sigma_{{a}}^{} \varepsilon_{{a}}^{{0}} \Delta \varepsilon_{{a}} \underset{\varepsilon _{a}^{0}}{\int} \varepsilon_{{a}}^{} \Bigl[ \varepsilon_{{a}}^{} \Bigr]_{{\varepsilon _{a}^{0}}}^{{\varepsilon _{a}^{0} + \Delta \varepsilon _{a}}} \left(\vphantom{ \varepsilon _{a}^{0} + \Delta \varepsilon _{a} }\right. \varepsilon_{{a}}^{{0}} \Delta \varepsilon_{{a}}^{} \left.\vphantom{ \varepsilon _{a}^{0} + \Delta \varepsilon _{a} }\right) \varepsilon_{{a}}^{{0}}$$ (16.23)

This can be written as:

$$\Delta \sigma_{{a}}^{} {\frac{{ E_{\mathrm{u}} \left( \varepsilon _{a}^{0} + \Delta \var... ...) - E_{\mathrm{u}}^{0} \, \varepsilon _{a}^{0} }}{{ \Delta \varepsilon _{a} }}} \bar{{E}}_{{\mathrm{ut}}}^{} \Delta \varepsilon_{{a}}^{}$$ (16.24)

This linear expression relating stress increment to strain increment is suitable for generalization and usage in DIANA. The averaged Young's modulus $\bar{{E}}_{{\mathrm{ut}}}^{}$ is found for the generalized case by substitution of the deviatoric strain invariant:

$$\bar{{E}}_{{\mathrm{ut}}}^{} {\frac{{ E_{\mathrm{u}} \, \varepsilon _{\mathrm{eq}} - E_{\mathr... ...m{eq}}^{0} }}{{ \varepsilon _{\mathrm{eq}} - \varepsilon _{\mathrm{eq}}^{0} }}}$$ (16.25)

In which $\varepsilon_{{\mathrm{eq}}}^{}$ is the updated equivalent strain and E_u the secant stiffness for this strain, obtained with (16.21). All variables in the right hand side of (16.25) are known in DIANA before the stress is updated. The average Young's modulus is substituted into the material stiffness matrix D , which in the general three-dimensional case results in:

$${\frac{{ \bar{E}_{\mathrm{ut}} }}{{ ( 1 + \nu ) ( 1 - 2 \nu ) }}} \left[\vphantom{ \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 ... ...llskipamount] 0 & 0 & 0 & 0 & 0 & \dfrac{ 1 - 2 \nu }{ 2 } \end{array} }\right. \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 & 0 \\ [\smallsk... ...\\ [\smallskipamount] 0 & 0 & 0 & 0 & 0 & \dfrac{ 1 - 2 \nu }{ 2 } \end{array} \left.\vphantom{ \begin{array}{cccccc} 1-\nu & \nu & \nu & 0 & 0 ... ...llskipamount] 0 & 0 & 0 & 0 & 0 & \dfrac{ 1 - 2 \nu }{ 2 } \end{array} }\right]$$ (16.26)

In which, for the Jardine model, Poisson's ratio $\nu$ is set to 0.49 by default. The stress increment in the generalized case is calculated with:

$$\Delta \boldsymbol\sigma \Delta \boldsymbol\varepsilon$$ (16.27)