Chapter 1
Overview of Continuum Mechanics

1.1 Definition of Tensor

1.1.1 Vectors and Tensors

In physics and mechanics, certain physical quantities are inherently independent of the choice of coordinate systems. A typical example is the vector, which represents a geometric quantity possessing both magnitude and direction. Common vector quantities in mechanics include displacement, velocity, force, and momentum, which are conventionally represented as directed line segments.

Vectors play a fundamental role across various natural sciences, including mathematics, physics, and engineering. Geometrically, a vector is often visualized as an arrowed line segment, where the length denotes its magnitude and the arrow indicates its direction. In contrast to vectors, scalar quantities-such as mass, temperature, and energy-possess magnitude but lack direction.

In mathematics, vectors are formally defined as elements of vector spaces (or linear spaces), which are abstract structures characterized by operations of addition and scalar multiplication. When interpreted physically, defining a vector requires the introduction of a norm and an inner product within the framework of Euclidean space, enabling the quantification of length and angle. For instance, the gradient of a scalar field is a vector field, while the derivative of a scalar with respect to another scalar remains a scalar.

Although a vector is an invariant geometric entity, its numerical components vary with the choice of coordinate system. To represent a vector, one must specify its components relative to a particular basis. These components transform according to specific rules-coordinate transformation laws-that preserve the underlying physical meaning of the vector. This transformation behavior underpins the vector's coordinate independence and leads naturally to the generalized concept of tensors.

Tensors generalize vectors and scalars, providing a powerful mathematical framework for formulating physical laws in a coordinate-independent manner. In many physical theories, it is necessary to express laws in terms of component quantities; however, the component values typically change under a transformation of the coordinate system. To ensure the validity of physical laws in any frame of reference, the transformation rules must apply consistently to all terms in an equation [1-3]. This principle guarantees the covariance of physical laws.

From a physical standpoint, the components of a tensor correspond to measurable quantities in a given frame of reference, whereas the tensor itself represents the intrinsic physical quantity. Observations may differ between reference frames, but the form of the physical laws and equations remains invariant. To fully describe a tensorial physical quantity, one must understand not only its components in a particular frame but also how those components transform under changes of reference. It is this transformation behavior that preserves the form of physical equations and ensures their universal applicability.

1.1.2 Definition of Tensors

1. Definitions

   The introduced coordinate system represents the physical quantity with some directional combination. Let the physical quantity be represented in different Cartesian coordinate systems as:

   (1.1)

   where is the combined base vector required to represent the directionality of the physical quantity, and let the number of combined base vectors be (i.e. base vectors multiplied). If its coordinate components meet the following coordinate conversion rules:

   (1.2)

   Combined Eq. (1.1) has

   (1.3)

   i.e. such a physical quantity is called a tensor, and the number of the combined basis vectors is called the order of this tensor. It can be seen that tensors represent physical quantities with a certain combination of directions, which are independent of the choice of coordinate system [4-7].

   For example, physical quantities such as temperature or mass density of an object are functions of position, independent of direction, and are called zero-order tensors, which are expressed as scalars. Other physical quantities are expressed by vectors, such as the displacement, velocity, acceleration of a particle, and the force exerted on an object, which can be expressed as , the number of combined base vectors , i.e. the vector is a first-order tensor. When the number of combined basis vectors is , it is called a second-order tensor, such as the inertia tensor representing the mass distribution of the object, the strain tensor representing the deformation at a point, and the corresponding stress tensor. Second-order tensors can be expressed as . A tensor can be represented in a variety of ways, such as in or other capitalized italic, bold. In the case of a given coordinate system, it can also be expressed in the form of omitting the base vector, as in the case of a second-order tensor , which represents a set of tensor components, where and are free indicators. The advantage of this representation is that it is expressed in scalar form, and when performing operations, it conforms to the well-known scalar algorithm [1, 5, 7]. Sometimes, for clarity, the second-order tensor can also be represented by a matrix of 3 × 3:

   (1.4)

   Something like represents a component of the tensor. Vectors can be represented in Ti or in arrays of 1 × 3:

   (1.5)
2. Objectivity of Tensors

   A tensor represents a physical quantity that is independent of the observer or the choice of reference frame. This invariance ensures that physical laws expressed in tensorial form remain valid in all coordinate systems, thereby satisfying the principle of objectivity required in physical formulations. Although the tensor itself is invariant, its component representation depends on the choice of basis vectors associated with a specific frame of reference. When the coordinate system changes, the basis vectors transform accordingly, and the tensor components adjust in such a way that the overall tensor remains unchanged. In this sense, a tensor is a covariant combination of basis and components. Thus, while the same tensor may have different component forms in different frames of reference, the underlying physical quantity it represents is preserved across all observers [1, 5, 7].

   Therefore, it can be understood that if the tensor satisfies the corresponding coordinate transformation law, the tensor can be objective. For example, the scalar a, the first-order tensor (vector) , the second-order tensor expressed in the reference line expressed under the reference line need to satisfy the following relationship:

   (1.6)

   So, at the level of coordinate transformation, the objectivity of the tensor and the indistinguishable property of the reference frame of the tensor are equivalent.

   However, it cannot be understood that if the tensor satisfies the coordinate transformation rules of the tensor, it is said that the tensor has objectivity, and there will be some special tensors. For example, the deformation gradient tensor is a second-order tensor categorically speaking, but its coordinate transformation rule is not , but , this is because is a special second-order tensor, also called a two-point tensor, because the base , the base under the current configuration changes with time and is affected by the rotation tensor ; At the same time, the substrate under the initial configuration is always unchanged, so it is not affected by . Therefore, the two-point tensor needs to satisfy the principle of objectivity, and the coordinate transformation law is the same as the first-order tensor, which is .

   At the same time, some tensors have objectivity, but their material time differentiation is not. For example, the Cauchy stress tensor has objectivity , but the Cauchy stress rate tensor is not:

   (1.7)

   Therefore, to ensure the objectivity of the Cauchy stress rate, several scholars have proposed objective stress rate formulations in tensorial form that are consistent with the transformation rules required for second-order tensors under a change of reference frame [8-10]. When establishing constitutive models-particularly those in rate form, such as hypoelastic models-it is essential to determine whether the employed tensor quantities are objective. If the material time derivative of a tensor is not objective, then any constitutive relation formulated using it may fail to be frame-indifferent, thereby violating a fundamental physical requirement.

   Objective constitutive equations are typically expressed in rate form to capture time-dependent behavior, and ensuring the objectivity of stress and strain rates is crucial for the correctness of these models. If non-objective tensors are used directly in the formulation, it becomes challenging to construct frame-indifferent constitutive laws without introducing additional correction terms or structural complexity.

   That said, it is not always necessary to use objective tensors in constitutive modeling. In certain cases, non-objective tensors can still yield physically meaningful results if the constitutive equations are constructed in terms of scalar invariants (e.g. norm or trace) or functions of tensors that remain invariant under observer transformations. In practice, many constitutive models adopt such an approach, leveraging observer-invariant quantities rather than explicitly ensuring the...