A two-surface thermomechanical plasticity model considering thermal cyclic behavior

Abstract

In thermal-related engineering such as thermal energy structures and nuclear waste disposal, it is essential to well understand volume change and excess pore water pressure buildup of soils under thermal cycles. However, most existing thermo-mechanical models can merely simulate one heating–cooling cycle and fail in capturing accumulation phenomenon due to multiple thermal cycles. In this study, a two-surface elasto-plastic model considering thermal cyclic behavior is proposed. This model is based on the bounding surface plasticity and progressive plasticity by introducing two yield surfaces and two loading yield limits. A dependency law is proposed by linking two loading yield limits with a thermal accumulation parameter n_c, allowing the thermal cyclic behavior to be taken into account. Parameter n_c controls the evolution rate of the inner loading yield limit approaching the loading yield limit following a thermal loading path. By extending the thermo-hydro-mechanical equations into the elastic–plastic state, the excess pore water pressure buildup of soil due to thermal cycles is also accounted. Then, thermal cycle tests on four fine-grained soils (natural Boom clay, Geneva clay, Bonny silt, and reconstituted Pontida clay) under different OCRs and stresses are simulated and compared. The results show that the proposed model can well describe both strain accumulation phenomenon and excess pore water pressure buildup of fine-grained soils under the effect of thermal cycles.

Appendices

Appendix 1

The consistency condition of loading surface in ACC2-T model is adopted herein:

$$\left( {\frac{{\partial f_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}} \right)^{t} :{\text{d}}{\mathbf{\sigma^{\prime}}} + \frac{{\partial f_{\text{I}} }}{\partial T}{\text{d}}T + \frac{{\partial f_{\text{I}} }}{{\partial \bar{p^{\prime}}_{{{\text{c}}0}} }}{\text{d}}\bar{p^{\prime}}_{{{\text{c}}0}} + \frac{{\partial f_{\text{I}} }}{{\partial r_{0} }}{\text{d}}r_{0} = 0$$

(18)

The plastic multiplier $\varLambda$ and the plastic modulus h are obtained in non-isothermal condition:

$$d{\varvec{\upvarepsilon}}^{\text{p}} = \varLambda \frac{{\partial g_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}$$

(19)

$$\varLambda = - \frac{{\left( {\frac{{\partial f_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}} \right)^{t} :{\text{d}}{\mathbf{\sigma^{\prime}}} + \frac{{\partial f_{\text{I}} }}{\partial T}{\text{d}}T}}{{\frac{{\partial f_{\text{I}} }}{{\partial \bar{p^{\prime}}_{{{\text{c}}0}} }}\frac{\upsilon }{\lambda - k}\bar{p^{\prime}}_{{{\text{c}}0}} \frac{{\partial g_{\text{I}} }}{{\partial p^{\prime}}} + \frac{{\partial f_{\text{I}} }}{{\partial r_{0} }}\frac{\upsilon }{\lambda - k}s(1 - r_{0} )\left( {\frac{{\partial g_{\text{I}} }}{{\partial p^{\prime}}} + A_{\text{d}} \frac{{\partial g_{\text{I}} }}{\partial q}} \right)}}$$

(20)

$$h = \frac{\upsilon }{\lambda - k}\left( {\frac{{\partial f_{\text{I}} }}{{\partial \bar{p^{\prime}}_{{{\text{c}}0}} }}\bar{p^{\prime}}_{{{\text{c}}0}} \frac{{\partial g_{\text{I}} }}{{\partial p^{\prime}}} + \frac{{\partial f_{\text{I}} }}{{\partial r_{0} }}s(1 - r_{0} )\left( {\frac{{\partial g_{\text{I}} }}{{\partial p^{\prime}}} + A_{\text{d}} \frac{{\partial g_{\text{I}} }}{\partial q}} \right)} \right)$$

(21)

The total strain increment is made up of mechanical elastic strain increment, thermal elastic strain increment and plastic strain increment.

$${\text{d}}{\varvec{\upvarepsilon}} = {\text{d}}{\varvec{\upvarepsilon}}_{{\sigma^{\prime}}}^{\text{e}} + {\text{d}}{\varvec{\upvarepsilon}}_{\text{T}}^{\text{e}} + {\text{d}}{\varvec{\upvarepsilon}}^{\text{p}}$$

(22)

The thermal elastic strain increment ${\text{d}}{\varvec{\upvarepsilon}}_{\text{T}}^{\text{e}}$ is defined as:

$${\text{d}}{\varvec{\upvarepsilon}}_{\text{T}}^{\text{e}} = - \frac{1}{3}{\mathbf{m}}\alpha_{1} {\text{d}}T$$

(23)

where m is the column vector with 1 at normal stress entries and 0 at shear stress entries.

By substituting Eq. (23) into Eq. (22), the general incremental stress–strain relation can be expressed:

$${\text{d}}{\mathbf{\sigma^{\prime}}} = {\mathbf{D}}_{\text{e}} {\text{d}}{\varvec{\upvarepsilon}}_{{\sigma^{\prime}}}^{\text{e}} = {\mathbf{D}}_{\text{e}} \left( {{\text{d}}{\varvec{\upvarepsilon}} + \frac{1}{3}{\mathbf{m}}\alpha_{1} {\text{d}}T - {\text{d}}{\varvec{\upvarepsilon}}^{\text{p}} } \right)$$

(24)

where D_e is the mechanical elastic matrix.

By substituting the plastic multiplier into Eq. (24), the differential stress–strain equations can be obtained:

$${\text{d}}{\mathbf{\sigma^{\prime}}} = {\mathbf{D}}_{\text{ep}} {\text{d}}{\varvec{\upvarepsilon}} + {\mathbf{D}}_{\text{et}} {\text{d}}T$$

(25)

where D_ep, D_et are the mechanical elasto-plastic matrix and the thermal elasto-plastic matrix, respectively.

$${\mathbf{D}}_{\text{ep}} = {\mathbf{D}}_{\text{e}} - \frac{{{\mathbf{D}}_{\text{e}} \frac{{\partial g_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}\left( {\frac{{\partial f_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}} \right)^{t} {\mathbf{D}}_{\text{e}} }}{{\left( {\frac{{\partial f_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}} \right)^{t} {\mathbf{D}}_{\text{e}} \frac{{\partial g_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}} + h}}$$

(26)

$${\mathbf{D}}_{\text{et}} = \frac{{{\mathbf{D}}_{\text{e}} \frac{{\partial g_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}\left[ { - \frac{1}{3}\left( {\frac{{\partial f_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}} \right)^{t} {\mathbf{D}}_{\text{e}} {\mathbf{m}}\alpha_{1} - \frac{{\partial f_{\text{I}} }}{\partial T}} \right]}}{{\left( {\frac{{\partial f_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}}} \right)^{t} {\mathbf{D}}_{\text{e}} \frac{{\partial g_{\text{I}} }}{{\partial {\mathbf{\sigma^{\prime}}}}} + h}} + \frac{1}{3}{\mathbf{D}}_{\text{e}} {\mathbf{m}}\alpha_{1}$$

(27)

Appendix 2

Assume the soil is loaded and heated from Point A to Point B under two different loading paths I ($A \to C \to B$) and II($A \to D \to B$) in Fig. 5. The soil is at the initial stress state ($p^{\prime}_{0} ,r_{0} ,p^{\prime}_{c0} ,T_{1}$) which automatically satisfies the yield condition:

$$f_{I0} = p^{\prime}_{0} - r_{0} p^{\prime}_{{{\text{c}}0}} \exp \left[ { - \alpha_{0} r_{0}^{{n_{\text{c}} }} \left( {T_{1} - T_{0} } \right)} \right] = 0$$

(28)

Under loading path I, the soil element is isotropically loaded to stress point $C$($p^{\prime}_{0} + \Delta p^{\prime}_{1} ,r_{1} ,p^{\prime}_{{{\text{c}}1}} ,T_{1}$) at constant temperature $T_{1}$.

$$f_{{{\text{I}}1}} = p^{\prime}_{0} { + }\Delta p^{\prime}_{1} - r_{1} p^{\prime}_{{{\text{c}}1}} \exp \left[ { - \alpha_{0} r_{1}^{{n_{\text{c}} }} \left( {T_{1} - T_{0} } \right)} \right] = 0$$

(29)

$$p^{\prime}_{{{\text{c}}1}} = p^{\prime}_{{{\text{c}}0}} \exp \left[ {\frac{\upsilon }{\lambda - \kappa }\Delta \varepsilon_{v1}^{\text{p}} } \right]$$

(30)

$$r_{1} = 1 + \left( {r_{0} - 1} \right)\exp \left[ { - \frac{\upsilon }{\lambda - \kappa }s\Delta \varepsilon_{{{\text{v}}1}}^{\text{p}} } \right]$$

(31)

Then the soil element is heated up to stress point $B$ ($p^{\prime}_{0} + \Delta p^{\prime}_{1} ,r_{2} ,p^{\prime}_{{{\text{c}}2}} ,T_{1} + \Delta T_{1}$) at constant mean effective stress $p^{\prime}_{0} + \Delta p^{\prime}_{1}$.

$$f_{I2} = p^{\prime}_{0} { + }\Delta p^{\prime}_{1} - r_{2} p^{\prime}_{{{\text{c}}2}} \exp \left[ { - \alpha_{0} r_{2}^{{n_{\text{c}} }} \left( {T_{1} + \Delta T_{1} - T_{0} } \right)} \right] = 0$$

(32)

$$p^{\prime}_{{{\text{c}}2}} = p^{\prime}_{{{\text{c}}1}} \exp \left[ {\frac{\upsilon }{\lambda - \kappa }\Delta \varepsilon_{{{\text{v}}2}}^{\text{p}} } \right] = p^{\prime}_{{{\text{c}}0}} \exp \left[ {\frac{\upsilon }{\lambda - \kappa }\left( {\Delta \varepsilon_{{{\text{v}}1}}^{\text{p}} + \Delta \varepsilon_{{{\text{v}}2}}^{\text{p}} } \right)} \right]$$

(33)

$$r_{2} = 1 + \left( {r_{1} - 1} \right)\exp \left[ { - \frac{\upsilon }{\lambda - \kappa }s\Delta \varepsilon_{{{\text{v}}2}}^{\text{p}} } \right] = 1 + \left( {r_{0} - 1} \right)\exp \left[ { - \frac{\upsilon }{\lambda - \kappa }s\left( {\Delta \varepsilon_{{{\text{v}}1}}^{\text{p}} + \Delta \varepsilon_{{{\text{v}}2}}^{\text{p}} } \right)} \right]$$

(34)

Under loading path II, the soil element is firstly heated up to stress point $D$($p^{\prime}_{0} ,r_{3} ,p^{\prime}_{{{\text{c}}3}} ,T_{1} + \Delta T_{1}$) at constant mean effective stress $p^{\prime}_{0}$.

$$f_{I 3} = p^{\prime}_{0} - r_{ 3} p^{\prime}_{\text{c3}} \exp \left[ { - \alpha_{0} r_{ 3}^{{n_{\text{c}} }} \left( {T_{1} { + }\Delta T_{1} - T_{0} } \right)} \right] = 0$$

(35)

$$p^{\prime}_{\text{c3}} = p^{\prime}_{{{\text{c}}0}} \exp \left[ {\frac{\upsilon }{\lambda - \kappa }\Delta \varepsilon_{\text{v3}}^{\text{p}} } \right]$$

(36)

$$r_{ 3} = 1 + \left( {r_{0} - 1} \right)\exp \left[ { - \frac{\upsilon }{\lambda - \kappa }s\Delta \varepsilon_{\text{v3}}^{\text{p}} } \right]$$

(37)

Then, the soil element is isotropically loaded to stress point $B$($p^{\prime}_{0} + \Delta p^{\prime}_{1} ,r_{ 4} ,p^{\prime}_{\text{c4}} ,T_{1} + \Delta T_{1}$) at constant temperature $T_{1} + \Delta T_{1}$.

$$f_{\text{I4}} = p^{\prime}_{0} { + }\Delta p^{\prime}_{1} - r_{ 4} p^{\prime}_{\text{c4}} \exp \left[ { - \alpha_{0} r_{ 4}^{{n_{\text{c}} }} \left( {T_{1} + \Delta T_{1} - T_{0} } \right)} \right] = 0$$

(38)

$$p^{\prime}_{\text{c4}} = p^{\prime}_{\text{c3}} \exp \left[ {\frac{\upsilon }{\lambda - \kappa }\Delta \varepsilon_{\text{v4}}^{\text{p}} } \right] = p^{\prime}_{{{\text{c}}0}} \exp \left[ {\frac{\upsilon }{\lambda - \kappa }\left( {\Delta \varepsilon_{\text{v3}}^{\text{p}} + \Delta \varepsilon_{\text{v4}}^{\text{p}} } \right)} \right]$$

(39)

$$r_{ 4} = 1 + \left( {r_{ 3} - 1} \right)\exp \left[ { - \frac{\upsilon }{\lambda - \kappa }s\Delta \varepsilon_{\text{v4}}^{\text{p}} } \right] = 1 + \left( {r_{0} - 1} \right)\exp \left[ { - \frac{\upsilon }{\lambda - \kappa }s\left( {\Delta \varepsilon_{\text{v3}}^{\text{p}} + \Delta \varepsilon_{\text{v4}}^{\text{p}} } \right)} \right]$$

(40)

Compare Eqs. (32)–(34) and Eqs. (38)–(40), it appears clearly that the two different loading paths would reach the same final stress state, namely:

$$p^{\prime}_{{{\text{c}}2}} = p^{\prime}_{\text{c4}}$$

(41)

$$r_{ 2} = r_{ 4}$$

(42)

$$\Delta \varepsilon_{v1}^{\text{p}} + \Delta \varepsilon_{v2}^{\text{p}} = \Delta \varepsilon_{v3}^{\text{p}} + \Delta \varepsilon_{v4}^{\text{p}}$$

(43)

No matter Path I or Path II is followed, the plastic volumetric strain will be equal, which means that plastic volumetric strain in ACC2-T model is also loading path independent.