1 Introduction

Analytical mechanics, such as Newtonian mechanics, is the most fundamental discipline of mechanics. The basic task is to continuously propose new solutions to problems in mechanics, physics, mechanical devices, aerospace, and engineering technology [1]. Research objects in analytical mechanics include particles, rigid bodies, and particle systems. A particle system, also known as a particle system, can contain an finite or infinite number of particles. Macroscopic objects. such as multi-rigid bodies, elastic objects, elastic–plastic objects, and continuum systems and their complexes, can all be regarded as particle systems. The research objects in analytical mechanics include particles, rigid bodies, and particle systems. A particle system, also known as a particle system, can contain a finite or an infinite number of particles. Macroscopic objects such as multi-rigid bodies, elastic objects, elastic–plastic objects, and continuum systems and their complexes, can all be regarded as particle systems. Both rigid and deformable objects can be studied by using analytical mechanics. Analytical mechanics uses generalized coordinates, puts forward ideal constraints, and studies the motion of macro-objects from the perspective of determining the energy of the system [2, 3]. Analytical mechanics are two basic principles of the constraint mechanics system: the virtual displacement principle and d'Alembert–Lagrange principle (also known as the general dynamic equilibrium and virtual displacement principle of dynamics). Using the virtual displacement principle, we can solve the static problems of the constrained object system by using the d’Alamber–Lagrange principle. We can further derive various forms of dynamic equations and solve the dynamic problems of the constrained dynamical system [4]. The concept of the virtual displacement of an object is proposed in analytical mechanics. The virtual displacement of an object must satisfy its constraints. It is a collection of infinitesimal displacements that may occur in the space region at a certain instant or position. An infinite number of virtual displacements can exist that are independent of the motion of the object. Two basic principles of analytical mechanics are based on this concept [5, 6]. We should point out that the basic principle given by analytical mechanics, which should be applicable to rigid bodies and deformed objects, while analytical mechanics only provides the virtual displacement principle of rigid objects and their combinations, which solves the static problems of rigid objects [2,3,4,5]. The application of the basic principle given previously in analytical mechanics is limited.

Macroscopic objects can be divided into rigid and deformed types. The static and dynamic problems of rigid bodies and their combinations have been comprehensively and deeply studied in existing theories of analytical mechanics, which has been formed [6, 7, 8]. Deformable objects can be divided into elastic and elastic–plastic objects. The virtual displacement principle of deformed objects has been described in structural mechanics, which is the basic principle of deformed objects [9, 10]. The virtual displacement principle of deformed objects is that the virtual work done directly by external forces (including volume force and area force) is equal to the strain energy required for deformed objects, or the virtual displacement principle is obtained using the second law of thermodynamics. However, in these studies, the important concepts of virtual displacement and ideal constraints were rarely discussed. In fact, the virtual displacement of the deformation object must satisfy the constraint condition, and the constraints are ideal constraints. Elastic–plastic objects are deformable objects that can be elastic or plastic in different environments. Elastoplasticity is a basic property of a material formation. The virtual displacement principle of an elastic–plastic objects is of great value in research on material formation. The establishment of the virtual displacement principle of elastic–plastic deformable body is also based on the fact that the virtual work of the external force on the deformable body is equal to the virtual strain energy of the elastic body. It should be noted that the virtual displacement principle of elastic–plastic objects must also be limited by ideal constraints. It can be seen that the virtual displacement principle of deformed objects in structural mechanics and elasticity rarely involves the concept of ideal constraints, nor is it based on the balance of deformed objects under external forces.

In the history of scientific development, people have always tried to unify the basic theories of mechanics, elastic mechanics, continuum mechanics, thermodynamics, electromagnetics, solids and deformed objects to better reveal the development laws of nature. At the turn of the twentieth century, attempts to unify the laws of mechanics and laws of reversible thermodynamics were made successfully made by Hadamard [11] for the thermal expansion of ideal gas. Efforts to unify the two laws have continued and were recently extended to irreversible thermodynamic laws [12, 13]. In 1973, Germain [14] extended the principle of virtual work in classical continuum mechanics for continua to multipolar media, called the principle of virtual power, which was further extended by Maugin to couple the thermo-mechanical field with an electromagnetic field [15]. At the end of the twentieth century, many studies on the irreversible thermodynamics, including irreversible thermodynamics were conducted by Maugin [16] and others have been summarized in a major research monograph. Green and Rivlin [17] initiated a unified procedure for constructing theories of the thermomechanics of multipolar media, which was initiated by Green and Rivlin [17] and then extended to include thermodynamics by Green and Naghdi [18, 19]. Attempts were made by Sieniutycz and Berry [20] to define a Lagrangian in an action-type integral for the deformation of the continuum with dissipation and by Maugin [21] to develop the analytical mechanics of dissipative materials. In all these major works except the one by Biot, the constitutive equations of the material must still be supplemented from the laws of thermodynamics or other postulates. In 2011, Pao et al. unified the principle of virtual power of thermomechanics for fluids and solids with dissipation [22]. In recent years, the principle of virtual displacement has been widely used in science and engineering technology and many good results have been achieved [23,24,25,26]. For a deformed object system, the key to solve the problem is to consider the physical properties (constitutive relations) of the object materials.

Recently, the author has conducted dynamic research on flexible robot objects, studying the Lie symmetry and conservation of snake-like robot systems [27], algebraic structures, and Poisson integration methods [28], as well as the Noether symmetry and Lie symmetry of a wall-climbing robot system and their corresponding conservation quantities [29, 30]. These studies used classical methods to divide the flexible body into many segments, then summed them up, calculated the system's de Lagrange function, further introduced the transformation Lie group, considered a certain symmetry property of the flexible object, and provided the motion law for the object. Research on these flexible problems essentially adopts the method of studying rigid bodies and multi-body systems, without considering the constitutive relationship (physical properties) between the stress and strain of flexible materials. The application of the research method in other problems is not very convenient, so the symmetry research on snake robots and wall climbing robots appears to be incomplete. This article adopts the basic method of analytical mechanics, considering the constitutive relationship between stress and strain of deformable objects (physical properties of materials), and proposes the virtual displacement principle and the d'Alamber–Lagrange principle of deformable objects, which are two basic principles of deformable objects. By utilizing these two fundamental principles, all dynamic problems of deformable objects can be solved. These two basic principles are not only applicable to deformable objects such as elastic, plastic, elastic–plastic, and flexible objects, but also to rigid objects [31,32,33,34,35,36,37].

The research in this paper is divided into the following parts: Sect. 1, introduction; Sect. 2, in this part, the generalized virtual displacement principle of deformed objects is proposed using the method of analytical mechanics; Sect. 3, the d'Alembert–Lagrange principle of deformed objects is obtained using the method of analytical mechanics; Sect. 4, several forms of virtual displacement principle of deformed object are given; Sect. 5, virtual displacement principle of deformed objects is applied in plane polar coordinates, space cylindrical and spherical coordinates; and Sect. 6, a simple conclusion is given.

2 Generalized Virtual Displacement Principle of the Deformation Objects

The generalized virtual displacement principle of a deformed object is the basic principle of the deformed object dynamics. It can solve the static problems of deformed objects, including the deformation problem when the system is in equilibrium or in uniform motion under the action of an external force. In this section, we only study the deformation problem in equilibrium, and the conclusion is suitable for the deformation problem in uniform motion.

2.1 Establishing of Virtual Displacement Principle

A deformed volume element ΔVi in the equilibrium state is affected by external forces including the volume force \({{\varvec{f}}}_i^V\) distributed in region V and the area force \({{\varvec{f}}}_i^S\) acting on the area ΔSi, the other are internal forces \({{\varvec{f}}}_i^j\) (namely gravitation between particles), elastic restoring force \({{\varvec{f}}}_i^I\) and constraint force \({{\varvec{R}}}_i\), the resultant force is equal to zero, namely

$${{\varvec{f}}}_i^V + {{\varvec{f}}}_i^S + {{\varvec{f}}}_i^i + {{\varvec{f}}}_i^I + {{\varvec{R}}}_i = 0.$$
(1)

At a certain time t, it is assumed that the deformed volume element has a virtual displacement \(\delta {{\varvec{r}}}_i\); then the virtual work of the forces on the deformed object is written in the form

$$\begin{aligned} \Delta W & = \sum_{i = 1}^N {{{\varvec{f}}}_i^V \cdot \delta {{\varvec{r}}}_i \Delta V_i } + \sum_{i = 1}^N {{{\varvec{f}}}_i^S \cdot \delta {{\varvec{r}}}_i \Delta S_i } + \sum_{i = 1}^N {{{\varvec{f}}}_i^j \cdot \delta {{\varvec{r}}}_i \Delta V_i } \\ & \quad + \sum_{i = 1}^N {{{\varvec{f}}}_i^I \cdot \delta {{\varvec{r}}}_i \Delta V_i } + \sum_{i = 1}^N {{{\varvec{R}}}_i \cdot \delta {{\varvec{r}}}_i \Delta S_i } = 0, \\ \end{aligned}$$
(2)

where ΔSi is the deformed area element.

Since the gravitation between particles is a pair of forces and reactions, so

$$\sum_{i = 1,j = 1}^N {{{\varvec{f}}}_i^j } \cdot \delta {{\varvec{r}}}_i \Delta V_i = 0,\quad {{\varvec{f}}}_i^j = {{\varvec{f}}}_{ij} + {{\varvec{f}}}_{ji} ,{\text{and }}{{\varvec{f}}}_{ji} = - {{\varvec{f}}}_{ij}$$
(3)

Assumption based on the concept of ideal constraints, one has

$$\sum_{i = 1}^N {{{\varvec{R}}}_i } \cdot \delta {{\varvec{r}}}_i \Delta V_i = 0.$$
(4)

Substitution of Eqs. (3), (4) into Eq. (2),we have

$$\Delta W = \sum_{i = 1}^N {{{\varvec{f}}}_i^V \cdot \delta {{\varvec{r}}}_i \Delta V_i } + \sum_{i = 1}^N {{{\varvec{f}}}_i^S \cdot \delta {{\varvec{r}}}_i \Delta S_i } + \sum_{i = 1}^N {{{\varvec{f}}}_i^I \cdot \delta {{\varvec{r}}}_i \Delta V_i } = 0,$$
(5)

Subsequently the virtual work of the deformable body under virtual displacement \(\delta {{\varvec{r}}}_i\) can be written as follows:

$$\delta W = \int_V {{{\varvec{f}}}^V \cdot \delta {{\varvec{r}}}{\text{d}}V} + \int_V {{{\varvec{f}}}^S \cdot \delta {{\varvec{r}}}{\text{d}}S} + \int_V {{{\varvec{f}}}^I \cdot \delta {{\varvec{r}}}{\text{d}}V} = 0.$$
(6)

It is well known that the relationship between the virtual work of elastic resilience and the virtual strain energy of the deformation object is as follows:

$$\delta U = - \int_V {f^I \cdot \delta {{\varvec{r}}}{\text{d}}V} = \int_V {\sigma_{ij} \delta \varepsilon_{ij} } {\text{d}}V$$
(7)

This article adopts Einstein's summation rule, which means that repeated subscripts indicate summation. Substitution of the Eq. (7) into the Eq. (6), one has

$$\delta U = \int_V {{{\varvec{f}}}^V } \cdot \delta {{\varvec{r}}}{\text{d}}V + \int_S {{{\varvec{f}}}^S } \cdot \delta {{\varvec{r}}}{\text{d}}S = \delta W,$$
(8)

or

$$\delta U = \int_V {\sigma_{ij} \delta \varepsilon_{ij} {\text{d}}V = \int_V {{{\varvec{f}}}^v \cdot \delta {{\varvec{r}}}} } {\text{d}}V + \int_S {{{\varvec{f}}}^s \cdot \delta {{\varvec{r}}}{\text{d}}S} = \int_V {\left( {{{\varvec{f}}}^V + \nabla \cdot {{\varvec{f}}}^S } \right) \cdot } \delta {{\varvec{r}}}{\text{d}}V.$$
(9)

The Eqs. (8) or (9) is called the generalized virtual displacement principle of the deformation object in equilibrium, where the volume force \({{\varvec{f}}}^V\) is a force that is proportional to the mass of each particle of a deformed object, such as gravity, magnetic force and \({{\varvec{f}}}^S\) the inertial force, reaction force, friction force, etc. We call this \({{\varvec{f}}}^V + \nabla \cdot {{\varvec{f}}}^S = {{\varvec{f}}}\) the equivalent volume force.

2.2 Formulation of Generalized Virtual Displacement Principle

Generalized Virtual Displacement Principle: The sufficient and necessary condition for the balance of an object is that the sum of the virtual work on the virtual displacement of the active force acting on the defamation object, under the ideal constraint, including the external force and the elastic restoring force, is equal to zero. That is the sum of the virtual works of the external force is equal to the strain energy of defamation object.

We proposed the virtual displacement principle of a def0rmation object, which is a generalized principle of body balance. Using this principle, we can obtain the virtual displacement principles of elastic objects, plastic objects, elastic–plastic objects and flexible objects, as well as the virtual displacement principles of the particle, rigid object, and multibody.

3 Several Forms on the Virtual Displacement Principle

In this section, we present several expressions for the virtual displacement principle of deformed objects.

3.1 Vector Form of the Virtual Displacement Principle

We call the Eqs. (8) and (9), the vector form of the virtual displacement principle for the deformation object.

3.2 Vector Form on the Virtual Displacement Principle

As shown in Fig. 1, a deformed object is an elastic body consisting of N flexible volume elements, that deform under the action of an external force. The inertial coordinate system is established at the bottom of the elastic body the unit vectors of the coordinate axis are (e1, e2, e3), and the other end is the free end.

Fig. 1
figure 1

Elastic body consist of N flexible volume elements

Full size image

In contrast to the rigid body system, the deformation of the body under the action of an external force and the relative position of each point in the system also changes. To accurately describe the motion and deformation of the body, a fixed inertial coordinate system (e1, e2 e3) and a moving coordinate system (b1, b2, b3) are established. The position vector for any point in the deformable body, is shown in Fig. 2.

Fig. 2
figure 2

Position vector representation

Full size image

For a deformed object, the radius vector \(\delta {{\varvec{r}}}\) of volume element is written as

$$\delta {{\varvec{r}}} = \delta \left( {{{\varvec{R}}} + A{{\varvec{x}}} + A{{\varvec{u}}}} \right) = \delta {{\varvec{X}}} + \delta \left( {A{{\varvec{u}}}} \right),\quad {{\varvec{X}}} = {{\varvec{R}}} + A{{\varvec{x}}}$$
(10)

where \(\delta {{\varvec{R}}}\) is the virtual displacement of the centroid of the deformed object, \(\delta {{\varvec{X}}}\) is the virtual displacement of the rigid body time, and \(\delta {{\varvec{u}}}\) is the virtual displacement of the deformed object, and A is a rotation transformation matrix from point P1 to point P2, which can be expressed as

$$A = \left[ {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {r_{13} } \\ {r_{21} } & {r_{22} } & {r_{23} } \\ {r_{31} } & {r_{32} } & {r_{33} } \\ \end{array} } \right],\quad {\text{then}}\,\left( {\begin{array}{*{20}c} {x_2 } \\ \begin{gathered} y_2 \hfill \\ z_2 \hfill \\ \end{gathered} \\ \end{array} } \right) = A\left( {\begin{array}{*{20}c} {x_1 } \\ \begin{gathered} y_1 \hfill \\ z_1 \hfill \\ \end{gathered} \\ \end{array} } \right){,}\,{\text{and}}\,AA^T = E$$
(11)

and A can be represented by the Euler angle(\(\alpha ,\beta ,\gamma\))

$$A = \left( {\begin{array}{*{20}c} {c_\alpha c_\beta } & {c_\alpha s_\beta s_\gamma - s_\alpha c_\gamma } & {c_\alpha s_\beta c_\gamma - s_\alpha s_\gamma } \\ {s_\alpha c_\beta } & {s_\alpha s_\beta s_\gamma - c_\alpha c_\gamma } & {s_\alpha s_\beta c_\gamma - c_\alpha s_\gamma } \\ { - s_\beta } & {c_\beta s_\gamma } & {c_\beta c_\gamma } \\ \end{array} } \right),{\text{here}}\quad c_i = \cos i,\;s_i = \sin i,\;i = \alpha ,\beta ,\gamma ,$$
(12)

A can also be represented by quaternion

$$A = \left( {\begin{array}{*{20}c} {\cos \theta + x^2 \left( {1 - \cos \theta } \right)} & {xy\left( {1 - \cos \theta } \right)} & {y\sin \theta } \\ { - xy\left( {1 - \cos \theta } \right)} & {\cos \theta + y^2 \left( {1 - \cos \theta } \right)} & { - x\sin \theta } \\ { - y\sin \theta } & {x\sin \theta } & {\cos \theta } \\ \end{array} } \right),\quad {\vec{\user2{q}}} = \left( {\theta ,x,y,z} \right)^T .$$
(13)

For convenience, the transformation matrix A can also be expressed in other form.

The Eq. (6) is expressed in the form

$$\begin{aligned} \delta W & = \int_V {{{\varvec{f}}}^V \cdot \delta {{\varvec{X}}}{\text{d}}V} + \int_S {{{\varvec{f}}}^S \cdot \delta {{\varvec{X}}}{\text{d}}S} \\ & \quad + \int_V {{{\varvec{f}}}^V \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}V} + \int_S {{{\varvec{f}}}^S \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}S} + \int_V {{{\varvec{f}}}^I \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}V} = 0, \\ \end{aligned}$$
(14)

It should be notes that Eq. (11) is the generalized virtual displacement principle of a deformable object in the equilibrium state. That is, the sum of the virtual work done by the volume area and elastic restoring force of the deformable object is equal to zero in the equilibrium state. At this time, the spatial motion of the deformable object is in a state of static or uniform motion, and the virtual displacement principle of a rigid object in the equilibrium state needs to be satisfied, that is,

$$\int_V {{{\varvec{f}}}^V \cdot \delta {{\varvec{X}}}{\text{d}}V} + \int_S {{{\varvec{f}}}^S \cdot \delta {{\varvec{X}}}{\text{d}}S} = 0.$$
(15)

Then, the deformation motion of the object must also be in a state of static or uniform motion, which satisfies the virtual displacement principle:

$$\int_V {{{\varvec{f}}}^V \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}V} + \int_S {{{\varvec{f}}}^S \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}S} + \int_V {{{\varvec{f}}}^I \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}V} = 0,$$
(16)

namely

$$\begin{aligned} & \int_V {{{\varvec{f}}}^V \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}V} + \int_S {{{\varvec{f}}}^S \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}S} \\ & \quad = \int_V {\left( {{{\varvec{f}}}^V + \nabla \cdot {{\varvec{f}}}^S } \right) \cdot \delta \left( {A{{\varvec{u}}}} \right){\text{d}}V = \int_V {{{\varvec{f}}} \cdot } } \delta \left( {A{{\varvec{u}}}} \right){\text{d}}V = \int_V {\sigma \delta \varepsilon {\text{d}}V} . \\ \end{aligned}$$
(17)

The Eq. (14) are called the vector forms of the generalized virtual displacement principle for the deformation objects. We also call the Eq. (17), the vector forms of the virtual displacement principle of the deformation objects and the Eq. (15), the virtual displacement principle of the particle system and rigid objects.

Now we give the following conclusions: for the deformed object in equilibrium or static state, its overall motion satisfies the virtual displacement principle (15) of rigid body, and the deformation motion satisfies the virtual displacement principle (17) of deformed object.

If \(\delta {{\varvec{u}}} = 0\), the Eq. (17) expresses the vector forms of the virtual displacement principle (15) for the rigid objects.

If we focus on the deformation of the objects in the equilibrium problem, Eq. (17) expresses the vector forms of the virtual displacement principle of the deformation objects.

3.3 Coordinate Component Form of Virtual Displacement Principle

Take the inertial coordinate system, the unit vectors of the coordinate axes are (e1, e2, e3), and we express the external forces \({{\varvec{f}}}^V\) and \({{\varvec{f}}}^S\), elastic restoring force \({{\varvec{f}}}^I\), and virtual displacement in the form of an inertial coordinate system. The external force \({{\varvec{f}}}^V ,f^{^S } ,f^I\) and \(\delta {{\varvec{r}}}\) are written in component forms

$$\begin{aligned} & {{\varvec{f}}}^V = f_x^V {{\varvec{e}}}_1 + f_y^V {{\varvec{e}}}_2 + f_z^V {{\varvec{e}}}_3 ,\quad {{\varvec{f}}}^S = f_x^S {{\varvec{e}}}_1 + f_y^S {{\varvec{e}}}_2 + f_z^S {{\varvec{e}}}_3 , \\ & {{\varvec{f}}}_i^I = f_x^I {{\varvec{e}}}_1 + f_y^I {{\varvec{e}}}_2 + f_z^I {{\varvec{e}}}_3 , \\ \end{aligned}$$
(18)

and

$$\begin{aligned} \delta {{\varvec{r}}} &= \delta \left( {A{{\varvec{x}}} + A{{\varvec{u}}}} \right) = \\ & \quad + A\delta x{{\varvec{e}}}_1 + A\delta y{{\varvec{e}}}_2 + A\delta z{{\varvec{e}}}_3 + A\delta u{{\varvec{e}}}_1 + A\delta v{{\varvec{e}}}_2 + A\delta w{{\varvec{e}}}_3 . \\ \end{aligned}$$
(19)

where A is a transformation matrix between two coordinate systems.

Then the virtual displacement principle is expressed the following coordinate component forms

$$\begin{aligned} & \int_V {A\left[ {f_x^V \delta x + f_y^V \delta y + f_z^V \delta z} \right]} {\text{d}}V \\ & \quad + \int_V {\left[ {\left( {f_x^V + f_x^I } \right)A\delta u + \left( {f_y^V + f_y^I } \right)A\delta v + \left( {f_z^V + f_z^I } \right)A\delta w} \right]} {\text{d}}V \\ & \quad + \int_S {\left( {f_x^S \delta x + f_y^S \delta y + f_z^S \delta z} \right)} {\text{d}}S + \int_S {\left( {f_x^S A\delta u + f_y^S A\delta v + f_z^S A\delta w} \right)} {\text{d}}S = 0, \\ \end{aligned}$$
(20)

namely

$$\int_V {\left[ {f_x^V \delta x + f_y^V \delta y + f_z^V \delta z} \right]} {\text{d}}V + \int_S {\left( {f_x^S \delta x + f_y^S \delta y + f_z^S \delta z} \right)} {\text{d}}S = 0$$
(21)
$$\begin{aligned} & \int_V {\left[ {\left( {f_x^V + f_x^I } \right)A\delta u + \left( {f_y^V + f_y^I } \right)A\delta v + \left( {f_z^V + f_z^I } \right)A\delta w} \right]} {\text{d}}V \\ & \quad + \int_S {\left( {f_x^S A\delta u + f_y^S A\delta v + f_z^S A\delta w} \right)} {\text{d}}S = 0. \\ \end{aligned}$$
(22)

the Eq. (22) is also written as

$$\begin{aligned} & \int_V {\left( {f_x^V A\delta u + f_y^V A\delta v + f_z^V A\delta w} \right)} {\text{d}}V \\ & \quad + \int_S {\left( {f_x^S A\delta u + f_y^S A\delta v + f_z^S A\delta w} \right)} {\text{d}}S = \int_V {\sigma_{ij} } \delta \varepsilon_{ij} {\text{d}}V. \\ \end{aligned}$$
(23)

We should point out Eq. (21) is the coordinate component form of the generalized virtual displacement principle of the deformed object considering the overall motion (rigid body) in the equilibrium state, and Eq. (23) is the coordinate component form of the generalized virtual displacement principle of the deformed object in the equilibrium state.

3.4 Generalized Coordinate Form of the Virtual Displacement Principle

The virtual displacement principle is the most common principle in statics and is widely used to solve static problems. The virtual displacement principle on generalized coordinates is expressed as follows. Under the double-sided, ideal, holonomic and stable constraints, the necessary and sufficient condition for the equilibrium of the deformation body is that the sum of the element work in any virtual displacement of the volume force, area force and elastic resilience acting on the system is equal to zero.

In the deformation body composed of N units, we take any unit body dVi and its area dSi, which is affected by the external force \({{\varvec{f}}}_i^V\), area force \({{\varvec{f}}}_i^S\), elastic restoring force \({{\varvec{f}}}_i^I\), and constraint force \({{\varvec{R}}}_i\).

Assuming that, the deformation body suffers l holonomic constraints, and the number of independent variables describing the system is n = 3N − l, then the system is described by n generalized coordinates q1, q2,\(\ldots\),qn. The position of a point on a deformed object is represented as

$${{\varvec{r}}} = {{\varvec{r}}}\left( {q_1 ,q_2 , \ldots ,q_n } \right),\quad {{\varvec{x}}}_i = {{\varvec{x}}}_i \left( {q_1 ,q_2 , \ldots ,q_n } \right),\quad i = 1,2,3\quad$$
(24)

The position of a point on a deformed object is represented. This deformation displacement can be expressed as

$${{\varvec{u}}}_i = {{\varvec{u}}}_i \left( {x_1 ({{\varvec{q}}}),x_2 ({{\varvec{q}}}),x_3 ({{\varvec{q}}})} \right) = {{\varvec{u}}}_i \left( {q_1 ,q_2 , \ldots ,q_n } \right),\quad i = 1,2,3$$
(25)

The stress and strain components can be written, respectively, as

$$\begin{aligned} & \sigma_{ij} = \sigma_{ij} \left( {x_1 ({{\varvec{q}}}),x_2 ({{\varvec{q}}}),x_3 ({{\varvec{q}}})} \right) = \sigma_{ij} \left( {q_1 ,q_2 , \ldots ,q_n } \right),\quad \\ & \varepsilon_{ij} = \varepsilon_{ij} \left( {x_1 ({{\varvec{q}}}),x_2 ({{\varvec{q}}}),x_3 ({{\varvec{q}}})} \right) = \varepsilon_{ij} \left( {q_1 ,q_2 , \ldots ,q_n } \right). \\ \end{aligned}$$
(26)

The components of the virtual displacement \(\delta {{\varvec{r}}}\),\(\delta {{\varvec{x}}}\) and \(\delta u\) are expressed, as follows:

$$\begin{aligned} \delta r & = \frac{\partial r}{{\partial q_s }}\delta q_s ,\quad s = 1,2, \ldots ,n \\ \delta x_1 & = \delta x = \frac{\partial x}{{\partial q_s }}\delta q_s ,\quad \delta x_2 = \delta y = \frac{\partial y}{{\partial q_s }}\delta q_s ,\quad \delta x_3 = \delta z = \frac{\partial z}{{\partial q_s }}\delta q_s , \\ \delta u & = \frac{\partial u}{{\partial x_i }}\frac{\partial x_i }{{\partial q_s }}\delta q_s = \frac{\partial u}{{\partial q_s }}\delta q_s ,\quad \delta v = \frac{\partial v}{{\partial x_i }}\frac{\partial x_i }{{\partial q_s }}\delta q_s = \frac{\partial v}{{\partial q_s }}\delta q_s ,\quad \delta w = \frac{\partial w}{{\partial x_i }}\frac{\partial x_i }{{\partial q_s }}\delta q_s = \frac{\partial w}{{\partial q_s }}\delta q_s , \\ \delta \varepsilon_{ij} & = \frac{{\partial \varepsilon_{ij} }}{\partial x_i }\frac{\partial x_i }{{\partial q_s }}\delta q_s = \frac{{\partial \varepsilon_{ij} }}{\partial q_s }\delta q_s ,\quad \delta \sigma_{ij} = \frac{{\partial \sigma_{ij} }}{\partial x_i }\frac{\partial x_i }{{\partial q_s }}\delta q_s = \frac{{\partial \sigma_{ij} }}{\partial q_s }\delta q\quad \left( {i,j = 1,2,3} \right) \\ \end{aligned}$$
(27)

The generalized virtual displacement principle of deformation objects are written in following general coordinate form

$$\begin{aligned} & \int_V {\sigma_{ij} } \frac{{\partial \varepsilon_{ij} }}{\partial q_s }\delta q_s {\text{d}}V = \int_V {\left( {f_x^V \frac{\partial x}{{\partial q_s }} + f_y^V \frac{\partial y}{{\partial q_s }} + f_z^V \frac{\partial z}{{\partial q_s }}} \right)} \delta q_s {\text{d}}V \\ & \quad + \int_S {\left( {f_x^S \frac{\partial x}{{\partial q_s }} + f_y^S \frac{\partial y}{{\partial q_s }} + f_z^S \frac{\partial z}{{\partial q_s }}} \right)\delta q_s } {\text{d}}S \\ & \quad + \int_V {\left( {f_x^V \frac{\partial u}{{\partial q_s }} + f_y^V \frac{\partial v}{{\partial q_s }} + f_z^V \frac{\partial w}{{\partial q_s }}} \right)\delta q_s {\text{d}}V} \\ & \quad + \int_S {\left( {f_x^S \frac{\partial u}{{\partial q_s }} + f_y^S \frac{\partial v}{{\partial q_s }} + f_z^S \frac{\partial w}{{\partial q_s }}} \right)\delta q_s } {\text{d}}S \\ & \quad = \int_V {Q_s^V \delta q_s } dV + \int_S {Q_s^S \delta q_s dS} + \int_V {\Phi_s^V \delta q_s dV} + \int_S {\Phi_s^S \delta q_s dS} . \\ \end{aligned}$$
(28)

where

$$\begin{aligned} Q_s^V & = f_x^V \frac{\partial x}{{\partial q_s }} + f_y^V \frac{\partial y}{{\partial q_s }} + f_z^V \frac{\partial z}{{\partial q_s }}, \\ Q_s^S & = f_x^S \frac{\partial x}{{\partial q_s }} + f_y^S \frac{\partial y}{{\partial q_s }} + f_z^S \frac{\partial z}{{\partial q_s }}, \\ \Phi_s^V & = f_x^V \frac{\partial u}{{\partial q_s }} + f_y^V \frac{\partial v}{{\partial q_s }} + f_z^V \frac{\partial w}{{\partial q_s }}, \\ \Phi_s^S & = f_x^S \frac{\partial u}{{\partial q_s }} + f_y^S \frac{\partial v}{{\partial q_s }} + f_z^S \frac{\partial w}{{\partial q_s }}. \\ \end{aligned}$$
(29)

When the deformed body is in equilibrium, we obtain the virtual displacement principle of rigidity objects as

$$\int_V {Q_s^V \delta q_s } dV + \int_S {Q_s^S \delta q_s } dS = 0,$$
(30)

and the virtual displacement principles of deformation objects

$$\int_V {\sigma_{ij} \frac{{\partial \varepsilon_{ij} }}{\partial q_s }} \delta q_s dV = \int_V {\Phi_s^V \delta q_s } dV + \int_S {\Phi_s^S \delta q_s } dS,$$
(31)

where \(\Phi_s^V\) and \(\Phi_s^S\) are the generalized volume and area forces, respectively. and the strain energy is expressed in the following form:

$$\begin{aligned} & \int_V {\sigma_{ij} } \delta \varepsilon_{ij} {\text{d}}V = \int_V {\sigma_{ij} } \frac{{\partial \varepsilon_{ij} }}{\partial q_s }\delta q_s {\text{d}}V = \int_V {\sigma_{ij} } \frac{{\partial \varepsilon_{ij} }}{\partial x_i }\frac{\partial x_i }{{\partial q_s }}\delta q_s {\text{d}}V, \\ & \quad i,j = 1,2,3;\quad s = 1,2, \ldots ,n. \\ \end{aligned}$$
(32)

We should show that the qs are lengths,\(\Phi_s^V\) and \(\Phi_s^S\) are the generalized forces, and the qs is angle, \(\Phi_s^V\), \(\Phi_s^S\), and generalized moments.

The Eqs. (19) and (20) are called the virtual displacement principles of deformed objects and rigid objects under the generalized coordinate respectively, and the Eq. (17), which is the generalized virtual displacement principle of the deformed objects in the generalized coordinate.

4 d'Alembert–Lagrange principle of defamation object

The d'Alembert principle extends Newton's law to a constrained particle system and further establishes the general dynamic equation of the particle system (d’Alembert–Lagrange principle), which is the basis of analytical mechanics and can solve the dynamic problems of particle systems and multi-body systems. In this section, the d'Alembert–Lagrange principle is discussed for deformable objects.

4.1 d'Alembert Principle of Defamation Object

4.1.1 d’Alembert Principle of the Particle System

The d'Alembert principle of the particle system has been gradually developed and improved by d'Alembert, Lagrange and Mei. More than 100 years after the principle was proposed, that is, in the half of the nineteenth century, people began to call the quantity ma the inertial force. The principle was interpreted as lows that after adding the inertial force F = ma, the sum of the vectors of the main force and inertial force of the particle was balanced with the binding force, that is,

$${{\varvec{F}}} + {{\varvec{F}}}^{\prime} + {{\varvec{N}}} = 0,$$
(33)

where F is denotes the main force (external force), N denotes the constraint force, and \({{\varvec{F}}}^{\prime} = - m{{\varvec{a}}}\) denotes the inertial force. For a particle system composed of N particles, the principle is expressed as follows: at any instant, the main force, constraint force and inertial force are in balance.

4.1.2 Generalized d'Alembert Principle of the Continuous Deformed Object

For a continuous deformed volume element ΔVi in the motion state, it is affected by external forces, including the volume force \({{\varvec{f}}}_i^V\) distributed in region V and the area force \({{\varvec{f}}}_i^S\) acting on the unit body area sugar ΔSi: the other are internal forces \({{\varvec{f}}}_i^i\) (namely gravitation between particles), elastic restoring force \({{\varvec{f}}}_i^I\) and constraint force \({{\varvec{R}}}_i\), and the object moves with an acceleration of a. Now we introduce the inertial force as

The continuous deformation element ΔVi is in equilibrium under the action of these forces: therefore, we have:

(34)

Then the continuous deformation system V in equilibrium, there force meet the following relationships.

(35)

Since the gravitation between particles is a pair of forces and reactions, so

$$\sum_{i = 1}^N {{{\varvec{f}}}_i^i } = 0,$$
(36)

then, the Eq. (35) is written as.

(37)

we call Eq. (37), the generalized d’Alembert principle of continuous deformed objects are expressed as follows: for a constrained continuous deformation object, at each instant, the volume force, area force, elastic restoring force, binding force and imaginary inertial force are in equilibrium. Equation (37) is the first expression of d'Alembert principle for a continuous deformation system.

The d’Alembert principle of continuous deformed objects can also be expressed as

$$\sum_{i = 1}^N {{{\varvec{f}}}_i^V } + \sum_{i = 1}^N {{{\varvec{f}}}_i^s } + \sum_{i = 1}^N {{{\varvec{f}}}_i^I } + \sum_{i = 1}^N {R_i } - \sum_{i = 1}^N {m{{\varvec{a}}}} = 0.$$
(38)

If the volume force, area force restraint force and elastic restoring force of the ith particle in the continuous deformation system are in equilibrium with the inertial force of the system, according to the principal vector of the equilibrium force system and the principal distance to any point A is zero, the following balance equation can be written.

(39)

where \(\sum_{i = 1}^N {f_i^{(e)} = } \sum_{i = 1}^N {{{\varvec{f}}}_i^V } + \sum_{i = 1}^N {{{\varvec{f}}}_i^s } + \sum_{i = 1}^N {{{\varvec{f}}}_i^I } + \sum_{i = 1}^N {R_i }\) denotes the join force of the external forces on the system. Equation (25) is the second expression of the d'Alembert principle for a continuous deformation system.

This was the result of the dynamic static method. Using Eq. (39), the dynamic problem of a continuous deformation system can be solved using the static method.

We should show that the dynamic static method developed from the d’Alembert principle of continuous deformation systems is theoretically equivalent to the theorem of momentum and momentum distance and can use the balance equation and problem-solving skills in statics in application.

4.2 Generalized d’Alembert–Lagrange Principle of Continuous Deformation Object

This section discusses the generalized d'Alembert–Lagrange principle of continuous deformation systems, including the establishment formulation and expression of the principle in several coordinates.

4.2.1 Establishment of Generalized d’Alembert–Lagrange Principle

The second expression of the generalized d’Alembert principle of a continuous deformation system is to summarize the dynamics problem as a balance problem under the action of all forces (including inertial forces) applied to the system. According to the generalized virtual displacement principle of continuous deformation systems, the necessary and sufficient conditions for system balance are the sum of the elementary work by all active forces acting on the double faced ideal constraint system at any virtual displacement equal to zero. The sum of the element work by all the active forces acting on the double faced ideal constraint system at any virtual displacement is equal to zero. This condition is applied to the moving continuous deformation system namely, the balance condition of the system under the action of the active force and inertial force is obtained in the following form

$$\Delta W = \sum_i^N {\left( {{{\varvec{f}}}}_i^V + {{\varvec{f}}}_i^s + {{\varvec{f}}}_i^I + {{\varvec{R}}}_i - m{{\varvec{{\ddot{r}}_i}}} \right) \cdot \delta {{\varvec{r}}}_i \Delta V_i = 0}$$
(40)

Assumption based on the concept of ideal constraints, one has.

$$\sum_{i = 1}^N {{{\varvec{R}}}_i } \cdot \delta {{\varvec{r}}}_i \Delta V_i = 0.$$
(41)

Substitution of the Eq. (41) into Eq. (40),we have

$$\Delta W = \sum_{i = 1}^N {{{\varvec{f}}}_i^V \cdot \delta {{\varvec{r}}}_i \Delta V_i } + \sum_{i = 1}^N {{{\varvec{f}}}_i^S \cdot \delta {{\varvec{r}}}_i \Delta S_i } + \sum_{i = 1}^N {{{\varvec{f}}}_i^I \cdot \delta {{\varvec{r}}}_i \Delta V_i } - \sum_i^N {m_i {\varvec{{r}}}_i \cdot \delta {{\varvec{r}}}_i \Delta {{\varvec{V}}}_i } = 0.$$
(42)

For the continuous deformation system, Eq. (42) can be written as

$$\delta W = \int_V {\left( {{{\varvec{f}}}^V + {{\varvec{f}}}^S + {{\varvec{f}}}^I - m{\varvec{{\ddot{r}}}}} \right)} \cdot \delta {{\varvec{r}}}{\text{d}}{{\varvec{V}}} = 0.$$
(43)

The relationship between the work of the elastic restoring force and the strain energy of the system is

$$\delta U = - \int_V {f^I \cdot \delta {{\varvec{r}}}{\text{d}}V} = \int_V {\sigma_{ij} \delta \varepsilon_{ij} } {\text{d}}V,$$
(44)

using the Eqs. (44) and (43), we have

$$\int_V {{{\varvec{f}}}^V \cdot \delta {{\varvec{r}}}{\text{d}}V + \int_S {{{\varvec{f}}}^S \cdot \delta {{\varvec{r}}}{\text{d}}S} - \int_V {m{\varvec{{\ddot{r}}}}} } \cdot \delta {{\varvec{r}}}{\text{d}}{{\varvec{V}}} = \int_V {{{\varvec{\sigma}}}\delta {{\varvec{\varepsilon}}}{\text{d}}V} ,$$
(45)

the Eq. (45) is called generalized d’Alembert–Lagrange principle of continuous deformation system.

4.2.2 Formulation of the d'Alembert–Lagrange Principle

The d'Alembert–Lagrange principle of continuous deformation object is described as follows:

For a continuous deformation object with double-sided constraints, the sum of the element work done by the active force (volume and area forces) acting on the system and the inertial force on any virtual displacement at the instantaneous position of the object is equal to the strain energy of the instantaneous object at each instantaneous motion.

The d’Alembert–Lagrange principle of a continuously deformed object is also brown as the general equation of dynamics and the virtual displacement principle of dynamics.

4.3 Several Coordinate Expressions of d'Alembert–Lagrange Principle for Continuously Deformed Object

4.3.1 Cartesian Coordinate Expression of the Principle

Principle (45) can be written the form of Cartesian coordinate

$$\int_V {\left( {f_x^V + \frac{\partial }{\partial x}f_x^S - m\ddot{x}} \right)\delta x{\text{d}}V} + \int_V {\left( {f_y^V + \frac{\partial }{\partial y}f_y^S - m\ddot{y}} \right)\delta y{\text{d}}V + \int_V {\left( {f_z^V + \frac{\partial }{\partial z}f_z^S - m\ddot{z}} \right)\delta z{\text{d}}V} } = \int_V {\sigma \delta \varepsilon } {\text{d}}V$$
(46)

4.3.2 Polar Coordinate Expression of Principle

In the polar coordinate system, the d'Alembert -Lagrange principle of continuous deformed object is expressed as

$$\int_V {\left( {\frac{\partial \sigma_r }{{\partial r}} + \frac{1}{r}\frac{{\partial \tau_{r\theta } }}{\partial \theta } + \frac{1}{r}\left( {\sigma_r - \sigma_\theta } \right) + f_r - m\ddot{r}} \right) \cdot \delta r{\text{d}}V} + \int_V {\left( {\frac{{\partial \tau_{r\theta } }}{\partial r} + \frac{1}{r}\frac{\partial \sigma_\theta }{{\partial \theta }} + \frac{{2\tau_{r\theta } }}{r} + f_\theta - m\ddot{\theta }} \right)} \cdot \delta \theta = 0$$
(47)

where, \(\sigma_r ,\sigma_\theta ,\tau_{r\theta }\) are stress tensors, which can be written as [8, 9]

$$\begin{aligned} & \sigma_r = \frac{1 - v^2 }{E}\left( {\varepsilon_r - \frac{v}{1 - v}\varepsilon_\theta } \right),\quad \sigma_\theta = \frac{1 - v^2 }{E}\left( {\varepsilon_\theta - \frac{v}{1 - v}\varepsilon_r } \right),\quad \tau_{r\theta } = \frac{{2\left( {1 + v} \right)}}{E}\gamma_{r\theta } , \\ & \varepsilon_r = \frac{\partial u_r }{{\partial r}},\quad \varepsilon_\theta = \frac{u_r }{r} + \frac{1}{r}\frac{\partial u_\theta }{{\partial \theta }},\quad \gamma_{r\theta } = \frac{1}{r}\frac{\partial u_r }{{\partial \theta }} + \frac{\partial u_\theta }{{\partial r}} - \frac{u_\theta }{r}. \\ \end{aligned}$$
(48)

4.3.3 Cylindrical Coordinate Form of the Principle

In the cylindrical coordinate system, principle (64) can be written as

$$\begin{aligned} & \int_V {\left( {\frac{\partial \sigma_r }{{\partial r}} + \frac{1}{r}\frac{{\partial \tau_{r\theta } }}{\partial \theta } + \frac{{\partial \tau_{zr} }}{\partial z} + \frac{\sigma_r - \sigma_\theta }{r} + f_r - m\ddot{r}} \right) \cdot \delta r{\text{d}}V} \\ & \quad + \int_V {(\frac{{\partial \tau_{r\theta } }}{\partial r} + \frac{1}{r}\frac{\partial \sigma_\theta }{{\partial \theta }} + \frac{{\partial \tau_{z\theta } }}{\partial z} + \frac{{2\tau_{r\theta } }}{r} + f_\theta } - m\ddot{\theta }) \cdot \delta \theta {\text{d}}V \\ & \quad + \int_V {\left( {\frac{{\partial \tau_{zr} }}{\partial r} + \frac{1}{r}\frac{{\partial \tau_{z\theta } }}{\partial \theta } + \frac{\partial \sigma_z }{{\partial z}} + \frac{{2\tau_{zr} }}{r} + f_z - m\ddot{z}} \right) \cdot \delta z{\text{d}}V} = 0, \\ & \sigma_r = \lambda \varphi + 2\mu \varepsilon_r ,\quad \sigma_\theta = \lambda \varphi + 2\mu \varepsilon_\theta ,\quad \sigma_z = \lambda \varphi + 2\mu \varepsilon_z , \\ & \tau_{zr} = \mu \gamma_{zr} ,\quad \tau_{r\theta } = \mu \gamma_{r\theta } ,\quad \tau_{z\theta } = \mu \gamma_{z\theta } , \\ \end{aligned}$$
(49)
$$\begin{gathered} \sigma_r = \lambda \varphi + 2\mu \varepsilon_r ,\quad \sigma_\theta = \lambda \varphi + 2\mu \varepsilon_\theta ,\quad \sigma_z = \lambda \varphi + 2\mu \varepsilon_z , \hfill \\ \tau_{zr} = \mu \gamma_{zr} ,\quad \tau_{r\theta } = \mu \gamma_{r\theta } ,\quad \tau_{z\theta } = \mu \gamma_{z\theta } , \hfill \\ \end{gathered}$$

The relationship between the strain and displacement components in generalized coordinates is as follows [8, 9]

$$\begin{aligned} & \varepsilon_r = \frac{\partial u_r }{{\partial r}},\quad \varepsilon_\theta = \frac{u_r }{r} + \frac{1}{r}\frac{\partial u_\theta }{{\partial \theta }},\quad \varepsilon_z = \frac{\partial w}{{\partial z}}, \\ & \gamma_{r\theta } = \frac{1}{r}\frac{\partial u_r }{{\partial \theta }} + \frac{\partial u_\theta }{{\partial r}} - \frac{u_\theta }{r},\quad \gamma_{z\theta } = \frac{1}{r}\frac{\partial w}{{\partial \theta }} + \frac{\partial u_\theta }{{\partial z}},\quad \gamma_{rz} = \frac{\partial u_r }{{\partial z}} + \frac{\partial w}{{\partial r}}, \\ \end{aligned}$$
(50)
$$\begin{aligned} & \sigma_r = \lambda \varphi + 2\mu \varepsilon_r ,\quad \sigma_\theta = \lambda \varphi + 2\mu \varepsilon_\theta ,\quad \sigma_z = \lambda \varphi + 2\mu \varepsilon_z , \\ & \tau_{zr} = \mu \gamma_{zr} ,\quad \tau_{r\theta } = \mu \gamma_{r\theta } ,\quad \tau_{z\theta } = \mu \gamma_{z\theta } , \\ & \lambda = \frac{Ev}{{\left( {1 + v} \right)\left( {1 - v} \right)}},\quad \mu = \frac{E}{{2\left( {1 + v} \right)}} \\ \end{aligned}$$
(51)

4.3.4 Spherical Coordinate Form of the Principle

In spherical coordinate elastic system, principle can be written in the form

$$\begin{aligned} & \int_V {\left( {\frac{\partial \sigma_r }{{\partial r}} + \frac{1}{r}\frac{{\partial \tau_{\theta r} }}{\partial \theta } + \frac{1}{r\sin \theta }\frac{{\partial \tau_{\varphi r} }}{\partial \varphi } + \frac{1}{r}\left( {2\sigma_r - \sigma_\theta - \sigma_\varphi + \tau_{r\theta } \cot \theta } \right) + f_r - m\ddot{r}} \right) \cdot \delta r{\text{d}}V} \\ & \quad + \int_V {(\frac{{\partial \tau_{r\theta } }}{\partial r} + \frac{1}{r}\frac{\partial \sigma_\theta }{{\partial \theta }} + \frac{1}{r\sin \theta }\frac{{\partial \tau_{\varphi \theta } }}{\partial \varphi } + \frac{1}{r}\left( {\left( {\sigma_\theta - \sigma_\varphi } \right)\cot \theta + 3\tau_{r\theta } } \right) + f_\theta } - m\ddot{\theta }) \cdot \delta \theta {\text{d}}V \\ & \quad + \int_V {\left( {\frac{{\partial \tau_{r\varphi } }}{\partial r} + \frac{1}{r}\frac{{\partial \tau_{\theta \varphi } }}{\partial \theta } + \frac{1}{r\sin \theta }\frac{\partial \sigma_\varphi }{{\partial \varphi }} + \frac{1}{r}\left( {3\tau_{r\varphi } + 2\tau_{\theta \varphi } \cot \theta } \right) + f_\varphi - m\ddot{\varphi }} \right) \cdot \delta \varphi {\text{d}}V} = 0, \\ \end{aligned}$$
(52)

where, \(\sigma_r ,\sigma_\theta ,\sigma_\varphi ,\tau_{r\theta } ,\tau_{\theta \varphi } ,\tau_{r\varphi }\) are given by [8, 9]

$$\begin{aligned} & \varepsilon_r = \frac{\partial u_r }{{\partial r}},\quad \varepsilon_\theta = \frac{u_r }{r} + \frac{1}{r}\frac{\partial u_\theta }{{\partial \theta }},\quad \varepsilon_\varphi = \frac{1}{r\sin \theta }\frac{\partial u_\varphi }{{\partial \varphi }} + \frac{u_\theta }{r}\cot \theta + \frac{u_r }{r}, \\ & \gamma_{r\theta } = \frac{1}{r}\frac{\partial u_r }{{\partial \theta }} + \frac{\partial u_\theta }{{\partial r}} - \frac{u_\theta }{r},\quad \gamma_{r\varphi } = \frac{1}{r\sin \theta }\frac{\partial u_r }{{\partial \varphi }} + \frac{\partial u_\varphi }{{\partial r}} - \frac{u_\varphi }{r}, \\ & \gamma_{\theta \varphi } = \frac{1}{r}\left( {\frac{\partial u_\varphi }{{\partial \theta }} - u_\varphi \cot \theta } \right) + \frac{1}{r\sin \theta }\frac{\partial u_\theta }{{\partial \varphi }}, \\ & \sigma_r = \frac{E}{1 + v}\left( {\frac{v}{1 - 2v}\varphi + \varepsilon_r } \right),\sigma_\theta = \frac{E}{1 + v}\left( {\frac{v}{1 - 2v}\theta + \varepsilon_\theta } \right),\sigma_\varphi = \frac{E}{1 + v}\left( {\frac{v}{1 - 2v}\theta + \varepsilon_\varphi } \right), \\ & \tau_{r\varphi } = \frac{E}{{2\left( {1 - 2v} \right)}}\gamma_{r\varphi } ,\tau_{r\theta } = \frac{E}{{2\left( {1 - 2v} \right)}}\gamma_{r\theta } ,\tau_{\theta \varphi } = \frac{E}{{2\left( {1 - 2v} \right)}}\gamma_{\theta \varphi } . \\ \end{aligned}$$
(53)

5 Generalized d’Alembert–Lagrange Principle of Elastic–Plastic Deformation Object

In this section, we study the generalized d’Alembert–Lagrange principle for elastic–plastic deformation objects.

5.1 The Stress–Strain and Constitutive Relationship of Ideal Elastic–Plastic Deformation Objects

For an ideal elastic–plastic deformation object, the form of the strain increments is expressed as

$$d\varepsilon_{ij} = d\varepsilon_{ij}^e + d\varepsilon_{ij}^p ,$$
(54)

where \(d\varepsilon_{ij}^e \;{\text{and}}\;d\varepsilon_{ij}^p\) represent the increment in elastic strain and plastic strain. The incremental constitutive relationships of the ideal elastic–plastic deformation object are

$$d\sigma_{ij} = C_{ijkl}^{ep} d\varepsilon_{kl} ,\quad C_{ij}^{ep} = C_{ijkl}^e + C_{ijkl}^p$$
(55)

or

$$d\sigma_{ij} = C_{ijkl} \left( {d\varepsilon_{kl} - d\lambda \frac{\partial g}{{\partial \varphi_{ij} }}} \right),\;d\lambda = \frac{1}{H}\frac{\partial f}{{\partial \sigma_{ij} }}C_{ijkl} d\varepsilon_{kl} ,\;H = \frac{\partial f}{{\partial \sigma_{ij} }}C_{ijkl} \frac{\partial g}{{\partial \sigma_{ijkl} }}.$$
(56)

where, \(f\left( {\sigma_{ij} } \right) = 0,\;g\left( {\sigma_{ijkl} } \right) = 0\) are the yield functions.

According to the Prandtl–Reuss (P–R)theory of ideal elastic–plastic materials, the increment in elastic strain deviation follows the generalized Hooke's law, and the increment in plastic strain deviation is proportional to the stress deviation, that is,

$$\begin{aligned} & de_{ij} = de_{ij}^e + de_{ij}^p ,\;de_{ij}^e = \frac{1}{2G}ds_{ij} ,\;de_{ij}^p = d\lambda s_{ij} , \\ & de_{ij} = \frac{1}{2G}ds_{ij} + d\lambda s_{ij} , \\ \end{aligned}$$
(57)

where \(d\lambda\) is a scaling factor that varies with the load, degree of deformation, and position of points.

The stress component satisfies Mises condition:\(\sigma_i = \sigma_s\), then

$$\begin{aligned} & de_{12}^p = de_1^p - de_2^p = d\lambda \left( {\sigma_1 - \sigma_2 } \right),\;de_{23}^p = de_2^p - de_3^p = d\lambda \left( {\sigma_2 - \sigma_3 } \right), \\ & de_{31}^p = de_3^p - de_1^p = d\lambda \left( {\sigma_3 - \sigma_1 } \right), \\ \end{aligned}$$
(58)

and we have

$$\sigma_i = \frac{3d\varepsilon_i^p }{{2d\lambda }}$$
(59)
$$d\varepsilon_i^p = \frac{{\sqrt {2} }}{3}\sqrt {{\left( {d\varepsilon_1^p - d\varepsilon_2^p } \right)^2 + \left( {d\varepsilon_2^p - d\varepsilon_3^p } \right)^2 + \left( {d\varepsilon_3^p - d\varepsilon_1^p } \right)^2 }} .$$
(60)

From Eq. (59), one has

$$d\lambda = \frac{3d\varepsilon_i^p }{{2\sigma_i }},$$
(61)

In other word, the proportion factor is related to the yield and deformation degrees of the material.

The P–R theory can be expressed in the form

$$de_{ij} = \frac{1}{2G}ds_{ij} + \frac{3d\varepsilon_i^p }{{2\sigma_i }}s_{ij} ,{\text{or }}de_{ij} = \frac{1}{2G}ds_{ij} + d\lambda s_{ij} ,$$
(62)

namely

$$\begin{gathered} de_x = \frac{1}{2G}ds_x + \frac{3d\varepsilon_i^p }{{2\sigma_i }}s_x ,{\text{and }}d\gamma_{xy} = \frac{1}{G}d\tau_{xy} + \frac{3d\varepsilon_i^p }{{2\sigma_i }}\tau_{xy} , \hfill \\ de_y = \frac{1}{2G}ds_y + \frac{3d\varepsilon_i^p }{{2\sigma_i }}s_y ,{\text{and }}d\gamma_{yz} = \frac{1}{G}d\tau_{yz} + \frac{3d\varepsilon_i^p }{{2\sigma_i }}\tau_{yz} , \hfill \\ de_z = \frac{1}{2G}ds_z + \frac{3d\varepsilon_i^p }{{2\sigma_i }}s_z ,{\text{and }}d\gamma_{zx} = \frac{1}{G}d\tau_{zx} + \frac{3d\varepsilon_i^p }{{2\sigma_i }}\tau_{zx} , \hfill \\ \end{gathered}$$
(63)

5.2 Prandtl-Reuss Theory for Representing Incremental Plastic Work

The plastic work increment of an ideal elastic–plastic material can be written as

$$dW = s_{ij} \cdot de_{ij} = \frac{1}{2G}s_{ij} \cdot ds_{ij} + d\lambda \cdot s_{ij} \cdot s_{ij}$$
(64)

The Mises condition can be used to obtain

$$dW = d\lambda \frac{2\sigma_s }{3},\;{\text{or }}\,d\lambda = \frac{3dW}{{2\sigma_s }},$$
(65)

So, the increment of strain deviation can be expressed as

$$de_{ij} = \frac{1}{2G}ds_{ij} + \frac{3dW}{{2\sigma_s }}s_{ij} ,$$
(66)
$$\begin{aligned} de_x & = \frac{1}{2G}ds_x + \frac{3dW}{{2\sigma_s }}s_x ,\quad d\gamma_{xy} = \frac{1}{G}d\tau_{xy} + \frac{3dW}{{2\sigma_s }}\tau_{xy} , \\ de_y & = \frac{1}{2G}ds_y + \frac{3dW}{{2\sigma_s }}s_y ,\quad d\gamma_{yz} = \frac{1}{G}d\tau_{yz} + \frac{3dW}{{2\sigma_s }}\tau_{yz} , \\ de_z & = \frac{1}{2G}ds_z + \frac{3dW}{{2\sigma_s }}s_z ,\quad d\gamma_{zx} = \frac{1}{G}d\tau_{zx} + \frac{3dW}{{2\sigma_s }}\tau_{zx} . \\ \end{aligned}$$
(67)

5.3 Levy–Mises Theory of Ideal Rigid Plastic Materials

The ideal rigid plastic mechanical model is

$$\sigma = \sigma_s ,$$
(68)

In the plastic zone, the elastic strain can be ignored, and the total strain is equal to the plastic strain, that is,

$$d\varepsilon_{ij} = d\varepsilon_{ij}^p ,\quad d\varepsilon_i = d\varepsilon_i^p ,\quad de_{ij} = de_{ij}^p ,$$
(69)

The stress component also satisfies the yield condition, and the L–M theory can be expressed as

$$d\varepsilon_{ij} = \frac{3d\varepsilon_i }{{2\sigma_s }}s_{ij} ,\quad d\lambda = \frac{3d\varepsilon_i^p }{{2\sigma_s }} = \frac{3d\varepsilon_i }{{2\sigma_s }},$$
(70)

namely,

$$\begin{aligned} & d\varepsilon_x = \frac{3d\varepsilon_i }{{2\sigma_s }}s_x ,\quad d\gamma_{xy} = \frac{3d\varepsilon_i }{{2\sigma_s }}\tau_{xy} , \\ & d\varepsilon_y = \frac{3d\varepsilon_i }{{2\sigma_s }}s_y ,\quad d\gamma_{yz} = \frac{3d\varepsilon_i }{{2\sigma_s }}\tau_{yz} , \\ & d\varepsilon_z = \frac{3d\varepsilon_i }{{2\sigma_s }}s_z ,\quad d\gamma_{zx} = \frac{3d\varepsilon_i }{{2\sigma_s }}\tau_{zx} . \\ \end{aligned}$$
(71)

5.4 The Deformation Theory (Hencky–Iliushin Theory) of Ideal Elastic–Plastic Deformation Objects

For a deformed object, the volume change is elastic and proportional to the average stress (i.e., plastic deformation with zero volume change), that is

$$\theta = \theta^e + \theta^p = \theta^e ,\quad \sigma_0 = K\theta ,\;K = \frac{E}{{3\left( {1 - 2\mu } \right)}}.$$
(72)

The stress deviation is directly proportional to the strain deviation; then the elastic stage: \(s_{ij} = 2Ge_{ij} ,\) in the plastic stage: \(s_{ij} = 2G^{\prime} e_{ij} ,\) here G′ is related to material properties and plastic deformation, and G′, is expressed as

$$G^{\prime} = \frac{\sigma_i }{{3\varepsilon_i }},$$
(73)

and

$$\begin{aligned} \sigma_i & = \sqrt {{\frac{1}{2}\left[ {\left( {\sigma_1 - \sigma_2 } \right)^2 + \left( {\sigma_2 - \sigma_3 } \right)^2 + \left( {\sigma_3 - \sigma_1 } \right)^2 } \right]}} , \\ \sigma_i & = 3G^{\prime} \frac{{\sqrt {2} }}{3}\sqrt {{\left( {\varepsilon_1 - \varepsilon_2 } \right)^2 + \left( {\varepsilon_2 - \varepsilon_3 } \right)^2 + \left( {\varepsilon_3 - \varepsilon_1 } \right)^2 }} = 3G^{\prime} \varepsilon_i . \\ \end{aligned}$$
(74)

Iliushin theory is expressed as

$$e_{ij} = \frac{3\varepsilon_i }{{2\sigma_i }}s_{ij} ,$$
(75)

that is

$$\begin{aligned} & e_x = \frac{3\varepsilon_i }{{2\sigma_i }}s_x ,\;\gamma_{xy} = \frac{3\varepsilon_i }{{\sigma_i }}\tau_{xy} ,e_y = \frac{3\varepsilon_i }{{2\sigma_i }}s_y ,\;\gamma_{yz} = \frac{3\varepsilon_i }{{\sigma_i }}\tau_{yz} , \\ & e_z = \frac{3\varepsilon_i }{{2\sigma_i }}s_z ,\;\gamma_{zx} = \frac{3\varepsilon_i }{{\sigma_i }}\tau_{zx} . \\ \end{aligned}$$
(76)

The principle indicates that at a certain instant, the volume strain of the system is proportional to the stress deviation.

The Hencky theory is expressed as

$$de_{ij} = \frac{1}{2G}ds_{ij} + d\lambda \cdot s_{ij} ,\;d\lambda = \frac{3d\varepsilon_i^p }{{2\sigma_i }}.$$
(77)

The deformation theory is also called the total quantity theory. The total quantity theory of elastic–plastic can be used to solve problems of proportional deformation and simple loading, as well as problems with a simple relationship between the stress intensity and strain intensity. For example

$$\begin{aligned} & \sigma = \varphi \left( \varepsilon \right), \to \sigma_i = \varphi \left( {\varepsilon_i } \right), \\ & \sigma = E\varepsilon , \to \sigma_i = E\varepsilon_i , \\ & \sigma = a\varepsilon^m , \to \sigma_i = a\varepsilon_i^m , \\ & \sigma = \sigma_s ,\sigma_i = \sigma_s . \\ \end{aligned}$$
(78)

5.5 The d’Alembert–Lagrange principle of an ideal elastic–plastic deformation object

Using the constitutive relationship of deformable materials, the d’Alembert–Lagrange principle of an ideal elastic–plastic deformation object can be expressed as

$$\int_V {{{\varvec{f}}}^V \cdot \delta {{\varvec{r}}}{\text{d}}V + \int_S {{{\varvec{f}}}^S \cdot \delta {{\varvec{r}}}{\text{d}}S} } = \int_V {{{\varvec{\sigma}}}\delta {{\varvec{\varepsilon}}}{\text{d}}V} .$$
(79)

where \({{\varvec{\sigma}}}\) and \(\delta {{\varvec{\varepsilon}}}\) are the stress and strain increments of the elastic–plastic deformation object, respectively.

5.6 Example

Thin-walled circular pipes made of incompressible elastic–plastic materials are subjected to axial tension and torque under Miese conditions find the stress components for \({{\varvec{\varepsilon}}} = \frac{\sigma_s }{E} = \frac{\sigma_s }{{3G}}\) and \(\gamma = = \frac{\sigma_s }{{\sqrt {3} G}}\) in the following three situations? and seek the work done?

  1. 1.

    First pull to \({{\varvec{\varepsilon}}} = \frac{\sigma_s }{{3G}}\), enter the plastic state, and then twist to \(\gamma = \frac{\sigma_s }{{\sqrt {3} G}}\);

  2. 2.

    First twist the thin cylinder to \(\gamma = = \frac{\sigma_s }{{\sqrt {3} G}}\) and enter the plastic state, and then pull it to \({{\varvec{\varepsilon}}} = \frac{\sigma_s }{{3G}}\);

  3. 3.

    Simultaneously pulling and twisting entering a plastic state.

Mises condition:

$$\sigma^2 + 3\tau^2 = \sigma_s^2$$
(80)
  1. (1)

    Solve:

First, the thin cylindrical cylinder is pull from the static to plastic state the strain components are \(\varepsilon_z :\, \to \frac{\sigma_s }{{3G}},\gamma = 0\), and the stress components are \(\sigma :\, \to \sigma_s ,\;\tau_z = 0\). In this process, the work is performed by torque \(\sigma\), which can be obtained using the d’Alembert–Lagrange principle (65) as:

$$W_1 = \int_0^{\sigma_s } \sigma d\left( {\frac{\sigma }{3G}} \right) = \frac{1}{6G}\sigma_s^2 = \frac{\sigma_s^2 }{{2E}}$$
(81)

Second, applying torque \(\tau\) in the plastic state causes tangential strain to enter a plastic state with a strain component of \(\gamma { = }\frac{\sigma_s }{{\sqrt {3} G}}\), and the stresses components are written as \(\sigma_z = \sigma ,\;\tau_{\theta z} = \tau ,\)\(\sigma_r = \sigma_\theta = \tau_{zr} = \tau_{r\theta } = 0,\)\(\sigma_0 = \frac{\sigma_z }{3},\) and \(s_z = \frac{2}{3}\sigma_s ,\;s_r = s_\theta = - \frac{\sigma_s }{3}\), respectively.

Owing to the incompressible volume of the system, the strain components of the thin cylindrical cylinder were \(\varepsilon_z = e_z = \frac{\sigma_s }{E},\;\varepsilon_r = e_r = - \frac{\sigma_s }{{2E}},\varepsilon_\theta = e_\theta = - \frac{\sigma_s }{{2E}},\;\gamma_{\theta z} = \frac{\sigma_s }{{\sqrt {3} G}} = \frac{{\sqrt {3} \sigma_s }}{E},\gamma_{\theta r} = \gamma_{zr} = 0\).

In addition, according to the increment of plastic work:

$$dW = s_{ij} de_{ij} = s_z de_z + s_r de_r + s_\theta de_\theta + \tau_{\theta z} d\gamma_{\theta z} + \tau_{zr} d\gamma_{zr} + \tau_{\theta r} d\gamma_{\theta r} = \sigma d\varepsilon + \tau d\gamma = \tau d\gamma .$$
(82)

The P–R theory represented by plastic work increment is written asthen, we have

$$de_{ij} = \frac{1}{2G}ds_{ij} + \frac{3dW}{{2\sigma_s }}s_{ij} ,$$
$$d\gamma_{ez} = \frac{1}{G}d\tau_{ez} + \frac{3dW}{{\sigma_s }}\tau_{ez} ,\;d\gamma = \frac{1}{G}d\tau + \frac{3\tau^2 }{{\sigma_s }}d\gamma ,$$
(83)

We can be obtain.

$$\tau = \frac{\sigma_s }{{\sqrt {3} }}th\frac{{\sqrt {2} G\gamma }}{\sigma_s },\;\sigma = \frac{\sigma_s }{{ch\frac{{\sqrt {2} G\gamma }}{\sigma_s }}},\;d\gamma = \frac{d\tau }{{G - \frac{3G\tau^2 }{{\sigma_s }}}}.$$
(84)

Using \(\gamma = \frac{\sigma_s }{{\sqrt {3} G}}\), one has.

$$\sigma = 0.648\sigma_s ,\;\tau = 0.439\sigma_s .$$
(85)

According to the d’Alembert–Lagrange principle, the work done by the external force and torque applied to an elastic–plastic thin cylinder can be written as

$$W_2 = \int {\tau d\gamma } = \int_0^\tau {\tau \frac{d\tau }{{G - \frac{3G\tau^2 }{{\sigma_s }}}}} = \frac{\sigma_s }{{6G}}\ln \left( {G - \frac{3G\tau^2 }{{\sigma_s }}} \right)\left| {_0^\tau } \right. = \frac{\sigma_s }{{6G}}\ln \left( {1 - 3 \times 0.439^2 \sigma_s } \right) = \ln \left( {0.521} \right)\frac{\sigma_s }{{6G}}.$$
(86)
  1. (2)

    Solve:

First, applying torque \(\tau\), in the plastic state causes tangential strain \(\gamma :0 \to \frac{\sigma_s }{{\sqrt {3} G}},\;\)\(\varepsilon = 0\), in the thin cylinder, and the stresses are \(\sigma = 0,\;\tau_{\theta z} :0 \to G\gamma_{\theta z} ,\)\(\tau_{\theta r} = \tau_{rz} = 0,\) where the work isperformed by torque \(\tau\), which can be obtained using the d’Alembert–Lagrange principle (79) as

$$W_1 = \int_0^{\sigma_s } {\frac{\sigma }{{\sqrt {3} }}} d\left( {\frac{\sigma }{{\sqrt {3} G}}} \right) = \frac{1}{6G}\sigma_s^2 = \frac{\sigma_s^2 }{{2E}}.$$
(87)

Second, applying pull force \(\sigma\), in the plastic state causes tangential strain entering a plastic state with a strain component of \(\varepsilon { = }\frac{\sigma_s }{{3G}}\), stresses components are written as \(\sigma_z = \sigma ,\;\tau_{\theta z} = \tau ,\)\(\sigma_r = \sigma_\theta = \tau_{zr} = \tau_{r\theta } = 0.\)

According to the increment of plastic work:

$$dW = s_{ij} de_{ij} = \sigma d\varepsilon + \tau d\gamma = \sigma d\varepsilon .$$
(88)

Due to, \(d\varepsilon = \frac{d\sigma }{{3G}} + \frac{\sigma^2 d\varepsilon }{{\sigma_s^2 }},\) namely \(d\varepsilon = \frac{d\sigma }{{3G - \frac{3G\sigma^2 }{{\sigma_s^2 }}}}\), then, we have

$$\varepsilon = \frac{\sigma_s }{{3G}}{\text{arcth}}\frac{\sigma }{\sigma_s } + c,$$
(89)

and using initial conditions, \(\varepsilon = 0.\;\sigma = 0,\) then c = 0, we have \(\sigma = \sigma_s {\text{th}}\frac{3G\varepsilon }{{\sigma_s }}\). When \(\varepsilon = \frac{\sigma_s }{{3G}},\) one has

$$\sigma = 0.762\sigma_s ,\tau = 0.374\sigma_s$$
(90)

According to the d’Alembert–Lagrange principle, the work done by the external force and torque applied to an elastic–plastic thin cylinder can be written as

$$W_2 = \int_0^\sigma {\sigma d\varepsilon } = \int_0^\sigma {\sigma \frac{d\sigma }{{3G - \frac{3G\sigma^2 }{{\sigma_s^2 }}}}} = \frac{\sigma_s^2 }{{6G}}\ln \left( {1 - \frac{\sigma^2 }{{\sigma_s^2 }}} \right)\left| {_0^{\sigma = 0.762\sigma_s } } \right. = \ln \left( {0.419} \right)\frac{\sigma_s^2 }{{6G}}$$
(91)
  1. (3)

    Solve:

When entering both tension and torsion states simultaneously, \(\varepsilon = \frac{\sigma_s }{{3G}},\)\(\gamma = = \frac{\sigma_s }{{\sqrt {3} G}}\), we have

$$\frac{\tau }{\sigma } = \frac{G\gamma }{{E\varepsilon }} = \frac{{\sqrt {3} }}{3},$$
(92)

and using Mises condition, we can obtain

$$\sigma = 0.707\sigma_s ,\tau = 0.408\sigma_s$$
(93)

According to the d’Alembert–Lagrange principle, the work done by the external force and torque applied to an elastic–plastic thin cylinder is

$$\begin{aligned} W & = \int_0^\sigma {\sigma d\varepsilon } + \int_0^\tau {\tau d\gamma } = \int_0^\tau {\tau \frac{d\tau }{{G - \frac{3G\tau^2 }{{\sigma_s }}}}} + \int_0^\sigma {\sigma \frac{d\sigma }{{3G - \frac{3G\sigma^2 }{{\sigma_s^2 }}}}} + \\ & = \frac{\sigma_s }{{6G}}\ln \left( {1 - \frac{3\tau^2 }{{\sigma_s }}} \right)\left| {_0^{\tau = 0.408\sigma_s } } \right. + \frac{\sigma_s^2 }{{6G}}\ln \left( {1 - \frac{\sigma^2 }{{\sigma_s^2 }}} \right)\left| {_0^{\sigma = 0.707\sigma_s } } \right. = \ln \left( {0.502} \right)\frac{\sigma_s^2 }{{6G}} + \ln \left( {0.5} \right)\frac{\sigma_s^2 }{{6G}} \approx \ln \left( {0.5} \right)\frac{\sigma_s^2 }{{3G}} \\ \end{aligned}$$
(94)

Now, we should indicate that for an ideal elastic–plastic thin cylinder under axial tension and torque, strain \(\varepsilon = {{\sigma_s } / {3G}}\) and shear strain \(\gamma = {{\sigma_s } / {\sqrt {3} G}}\) occur. The effects of tension and torque on the work performed in (86), (91), and (94) in the plastic state under three conditions: pulling before twisting, twisting before pulling, and simultaneously pulling and twisting. In other word, the work done by an external force in a plastic state is not only related to the magnitude of the external force and torque but also to the order of action.

6 Conclusions

In the study, two basic principles of deformed objects were studied using analytical mechanics. The following conclusions were drawn:

  1. 1.

    The principle of virtual displacement of deformed objects in an equilibrium state proposed in elasticity and material mechanics is derived based on the functional relationship of energy conservation or the first law of thermodynamics, which article introduces virtual displacement and ideal constraints, calculates the virtual work done by external forces on a deformed object system and the strain energy of the deformed object, and proposes the principle of virtual displacement of the deformed object in an equilibrium state. The principle of virtual displacement is known as the principle of virtual work. The principle of the virtual displacement of deformable objects can solve all static problems of deformable objects.

  2. 2.

    Introducing inertial force as an external force in elasticity and elasticity, and proposing the virtual displacement principle (virtual work principle) for deformable objects in motion. This article introduces the inertial force as an external force proposes the d'Alembert principle, and uses the principle of virtual displacement to propose the d'Alamber–Lagrange principle for deformable objects, also known as the general dynamic equation for deformable objects. The d'Alembert Lagrangian principle for deformable objects can solve all dynamic problems of deformable objects.

  3. 3.

    The d'Alembert–Lagrangian principle for deformable objects proposed in this study is general and not only applicable to deformable objects (elastic objects, plastic objects, and elastic–plastic objects), but also to particles, rigid bodies, and particle systems.

  4. 4.

    This study unifies the two basic principles of the virtual displacement principle and the d'Alambert–Lagrange principle in analytical mechanics, elastic mechanics, and material mechanics.

  5. 5.

    The use of the d'Alembert–Lagrangian principle presented in this article can conveniently derive Lagrangian equation and Hamiltonian equations for deformable objects.

  6. 6.

    This studt investigates the dynamic problems of thin elastic–plastic cylinders in polar coordinate systems, cylindrical coordinate systems, spherical coordinate systems, and elastic–plastic thin cylinders using the principle of d'Alembert–Lagrange principle.

  7. 7.

    Basic theories, dynamic equations, and important conclusions in analytical mechanics can be extended and applied to elastic mechanics, material mechanics, and general deformable object systems.

  8. 8.

    The use of the d'Alembert Lagrangian principle presented in this article can conveniently derive Lagrangian and Hamiltonian equations for deformable objects.

  9. 9.

    This article applies the d'Alamber–Lagrange principle to study the dynamic problems of deformable objects in polar, cylindrical, and spherical coordinate systems. It also investigates the constitutive relationship and strain energy of elastic–plastic thin cylinders under external forces and torques.

  10. 10.

    Basic theories, dynamic equations, and important conclusions in analytical mechanics can be extended and applied to elastic mechanics, material mechanics, and general deformable object systems.