2.1 Energy conversion analysis
According to the principle of conservation of energy, the process of external work on the sample is accompanied by energy conversion; when the material is an ideal brittle rock, the external work is completely converted into elastic strain energy and stored within the material. The digested energy is to overcome the rock's intrinsic cohesion in the phase of crack evolution, which means there is no extra energy dissipation until rock strength reaches the peak[10].
Due to the heterogeneity of rock materials, not all external work is converted into elastic strain energy and stored inside the material, and it is accompanied by other forms of energy dissipation, such as elastic strain energy[28], acoustic emission[29–31] that releases energy in the form of elastic waves phenomenon: deformation energy[32], which causes the internal state of the material to change, thermal energy[33], and so on. This portion of the energy is primarily used to reconstruct the internal stress field distribution, as well as to generate and decimate micro-cracks, which have been observed by the acoustic emission study of rock specimens under uniaxial loading[23, 26, 29–32, 34, 35].
The uniaxial compression process is accompanied by a certain level of AE ringing count events before approximately 50% of compressive strength, and the occurrence of the events is random[10]. Xie Heping[23] assumed that there is no heat exchange between the rock and the outside in the process, according to the first law of thermodynamics, its form is given in the following:
$${U^e}=\frac{1}{2}({\sigma _1}{\varepsilon _1}+{\sigma _2}{\varepsilon _2}+{\sigma _3}{\varepsilon _3})=\frac{1}{{2E}}[\sigma _{1}^{2}+\sigma _{2}^{2}+\sigma _{3}^{2} - 2\nu ({\sigma _1}{\text{ }}{\sigma _2}+{\sigma _1}{\sigma _3}+{\sigma _3}{\sigma _2})]$$
2
where U is the total energy density of external workmanship, Ue is the elastic strain energy density, and Ud is the dissipated energy density.
The one characteristic curve of uniaxial compression stress and strain obtained, shown in Fig. 2, is used as an example to illustrate the energy evolution of the compacted and the elastic deformation stage before the crack initiation stress threshold.
2.2 Measures of elastic strain energy density
Four assumptions are considered in this paper, that: (i) the rock material is perfectly elastic (no energy dissipation in elastic deformation and all elastic strain energy released after force removed) ; (ii) define the unit volume V and the unit has a closed surface S, \(\delta W\)and \(\delta U\)are the work of the external force and the increment of internal energy for unit volume, respectively; (iii) the external force is applied constantly, regardless of the effect of strain rate, to ensure that the unit body is in equilibrium at any time;(iv) the kinetic energy changes are ignored; These are guaranteed to be a quasi-static loading in uniaxial compression.
Based on the above assumptions, the stress tensor is decomposed into spherical and deviatoric stress tensors; the strain tensor is decomposed into spherical and deviatoric strain tensors, that is, strain energy is decomposed into the energy of volume \({W_1}\)and the energy of shape change\({W_2}\), the generalized form given in the following, respectively.
$${W_1}=\iiint_{V} {\left( {{X_u}+{Y_v}+{Z_w}} \right)}dV$$
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$${W_2}=\iint_{S} {\left( {{X^*}u+{Y^*}v+{Z^*}w} \right)dS}$$
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Where,,defined as the external force on per unit volume element, \({X^*}\),\({Y^*}\),\({Z^*}\) are the internal force on the surface of the unit element, the external force work W is entirely transformed into elastic strain energy and is defined as:
$$W={W_1}+{W_2}=\iiint_{V} {\left( {{X_u}+{Y_v}+{Z_w}} \right)}dV+\iint_{S} {\left( {{X^*}u+{Y^*}v+{Z^*}w} \right)dS}$$
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The work done by the external force acting on the unit body to produce a small displacement\(\delta W\), is expressed as:
$$\delta W=\delta {W_1}+\delta {W_2}=\iiint_{V} {{X_i}}\delta {u_i}dV+\iint_{S} {X_{i}^{*}}\delta {u_i}dS$$
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After substituting into the balance equation and boundary conditions, it is known from the divergence theorem:
$$\delta W=\iiint_{V} {{\sigma _{ij}}}\delta {u_{i,j}}dV$$
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The increase of internal energy is found:
$$\delta U=\iiint_{V} {{\sigma _{ij}}}\delta {\varepsilon _{i,j}}dV$$
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Known from Green’s formula, and let:
$$~~\frac{{\partial {u_0}\left( {{\varepsilon _i}_{j}} \right)}}{{\partial {\varepsilon _i}_{j}}}={\sigma _{ij}}$$
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Integrate the formula can easily find:
$$\int_{0}^{{{u_0}\left( {{\varepsilon _{ij}}} \right)}} {d{u_0}} ={u_0}\left( {{\varepsilon _{ij}}} \right) - {u_0}\left( 0 \right)$$
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where\({u_0}\left( {{\varepsilon _{ij}}} \right),{u_0}\left( 0 \right)\) respectively indicate the elastic strain energy density after loading and before loading.
Let\({u_0}\left( 0 \right)=0\), and the strain energy density is determined by its stress and strain increase, which can be defined as:
$${u_0}({\varepsilon _i}_{j})=\int_{0}^{{{\varepsilon _{ij}}}} {{\sigma _{ij}}} d{\varepsilon _{ij}}$$
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The unit volume strain energy density of a rock sample can be expressed as:
$$\frac{{dW}}{{dV}}=\int_{0}^{{{\varepsilon _{ij}}}} {{\sigma _{ij}}} d{\varepsilon _{ij}}$$
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2.3 Characteristics of strain energy evolution and initiation energy
According to Qingbin Meng et al.[28] energy self-inhibition evolution model and Zheng Zaisheng's [36] research on the nonlinear fitting formula of energy transfer in rock deformation under uniaxial compression with a certain stress level\(\sigma\). The rate of increment of the accumulated strain energy density was given[27, 34, 37] by:
$$\frac{1}{{{u^i}}}\cdot \frac{{d{u^i}}}{{d\sigma }}={a_i}({u^c}^{i} - {u^i}^{0}) - {b_i}{u^i}$$
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The formula \({u^c}^{i}\)represents the energy density converted from external input driving energy, \({u^i}^{0}\) is the lowest input strain energy density threshold, and, \({a_i}\),\({b_i}\) is the coefficient, which respectively reflects the degree of energy conversion and inhibition.
Integrate \({u^i}\) and find:
$${u^i}=\frac{{{a_i}({u^c}^{i} - {u^i}^{0}){e^{{a_i}}}{{^{{({u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{\sigma }}}{{{c_1}{b_i}[1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{\sigma }]}}$$
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with\({c_1}\),the integral constant.
Based on elasticity modulus\({E_i}\) and formula Eq. 13, find:
$${E_i}=\frac{{{\sigma ^2}}}{{{u^i}}}$$
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Derivative and let the first derivative be 0 as follows:
$$\frac{{d{E_i}}}{{d\sigma }}=\frac{{2\sigma {u^i} - {{\left( {{u^i}} \right)}^\prime }{\sigma ^2}}}{{{{\left( {{u^i}} \right)}^2}}}=0$$
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Eq. 14 substituted and organized, can find:
$${\sigma _i}=\frac{2}{{{a_i}\left( {{u^c}^{i} - {u^i}^{0}} \right) - {b_i}{u^i}}}=\frac{2}{{{a_i}\left( {{u^c}^{i} - {u^i}^{0}} \right) - \frac{{{a_i}({u^c}^{i} - {u^i}^{0}){e^{{a_i}}}{{^{{({u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\sigma _i}}}}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\sigma _i}}}}}}}=\frac{{2{a_i}\left( {{u^c}^{i} - {u^i}^{0}} \right)}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\sigma _i}}}}}$$
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After being simplified, when the elastic modulus reaches the maximum value, the strain energy accumulated\({u_i}\)and Stress\({\sigma _i}\)are obtained:
$${u_i}=c{e^{ - \frac{1}{2}}}{\sigma _i}$$
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And
$${\sigma _i}=\frac{{2{a_i}\left( {{u^c}^{i} - {u^i}^{0}} \right)}}{{1+{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{c/{b_i}}}{e^{{a_i}(}}{{^{{{u^c}^{i}+{u^i}^{0}}}}^)}^{{{\sigma _i}}}}}$$
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with the integral constant.
2.4 Energy digestion index (EDI)
From the concept of strain energy density, with isothermal conditions, the unit volume strain energy density can be expressed as:
$$\frac{{dW}}{{dV}}=\int_{0}^{{{\varepsilon _{ij}}}} {{\sigma _{ij}}} d{\varepsilon _{ij}}$$
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The above formula shows that the strain energy density of a unit can be determined by the stress and strain of the unit. The elastic stage of the stress-strain curve obtained from the experiment is partially enlarged, as shown in Fig. 3.
When the rock sample is loaded to point B in the uniaxial compression test, the elastic strain reaches a maximum, and the rock cracks. The area AOBD can be used to express the external force input energy during this process. The sample material is generally thought to be in the elastic stage at this point[35, 38, 39]. When the load is unloaded at point B, the formed unloading curve essentially coincides with the loading curve, indicating that no new crack damage is generated as a result of external force and no energy is dissipated[40–42].
The strain energy digested by the rock sample is represented by the area AODBC. Area ACB represents the portion of the external force's work that has not been converted into strain energy. At this point, the energy generated by the external force can be divided into two parts: the strain energy \({U^e}\)converted and digested by the rock sample, and the energy loss \({U^a}\) the external force work cannot be converted into strain energy. The rock material's energy digestion index Eq. 20 is established by analyzing the proportion of the energy digested in the total input energy. This formula is used to characterize the degree of response of the rock material to external loads and reflect the properties of the material itself. The energy digestion index EDI can be expressed as:
$$D=\frac{{{U^e}}}{U}=\frac{{{U^e}}}{{{U^a}+{U^e}}} \times 100\%$$
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The above formula shows that when a rock material is subjected to an external load, the energy generated by the external force is not completely converted into elastic strain energy and stored inside the material due to differences in the rock material with microstructure and ability to withstand the outside; in this process, the mechanical properties of rock materials are characterized by the energy conversion and digestion ability.