In this blog post, we are going to discover how the application of Taylor series aids in understanding the complexities of large deformations within nonlinear finite element analysis.
Introduction
In finite element analysis, we encounter various types of nonlinearities that can significantly affect our models:
Material Nonlinearity
This occurs when materials exhibit behaviors like plastic strain, common in metals such as steel, where strain hardening leads to a nonlinear stress-strain relationship post-yield strength.
Geometrical Nonlinearity
This arises when displacements become substantial, as seen in examples like a flexing fishing rod where traditional linear assumptions break down.
Contact Nonlinearity
Frictional contacts introduce nonlinearities, acting as new boundary conditions during the analysis altering the stiffness matrix. This results in a nonlinear relationship between force and displacement, especially evident during contact initiation.
Large Deformations
When structures undergo significant deformation, such as excessive stretching of a bar, factors like Poisson’s ratio cause the cross-sectional area to decrease considerably. Therefore, it becomes crucial to consider the instantaneous area rather than the initial area to accurately capture these large deformations.
In our upcoming blog post, we will delve into the formulation of expressions for both linear and angular strains, tailored specifically for handling large deformations within finite element analysis.
Deformation Calculation for Small Deformations – Linear FEA
In linear finite element analysis, we rely on the following expressions to evaluate both linear and angular (shear) strain.
Linear Deformation – X Direction
\varepsilon_x=\frac{\partial u}{\partial x}
Linear Deformation – Y Direction
\varepsilon_y=\frac{\partial v}{\partial y}
Linear Deformation – Z Direction
\varepsilon_z=\frac{\partial w}{\partial z}
Angular Deformation – XY Direction
\gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}
Angular Deformation – XZ Direction
\gamma_{xz}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}
Angular Deformation – YZ Direction
\gamma_{yz}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}
Deformation Calculation for Large Deformations – Nonlinear FEA
In nonlinear finite element analysis involving large deformations, we utilize the following expressions to quantify linear and angular strain:
Linear Deformation – X Direction
\varepsilon_x=\frac{\partial u}{\partial x}+\frac12\left[\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\right]
Linear Deformation – Y Direction
\varepsilon_y=\frac{\partial v}{\partial y}+\frac12\left[\left(\frac{\partial u}{\partial y}\right)^2+\left(\frac{\partial v}{\partial y}\right)^2+\left(\frac{\partial w}{\partial y}\right)^2\right]
Linear Deformation – Z Direction
\varepsilon_z=\frac{\partial w}{\partial z}+\frac12\left[\left(\frac{\partial u}{\partial z}\right)^2+\left(\frac{\partial v}{\partial z}\right)^2+\left(\frac{\partial w}{\partial z}\right)^2\right]
Angular Deformation – XY Direction
\gamma_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}+\frac{\partial u}{\partial x}\cdot\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\cdot\frac{\partial v}{\partial y}+\frac{\partial w}{\partial x}\cdot\frac{\partial w}{\partial y}
Angular Deformation – XZ Direction
\gamma_{xz}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}+\frac{\partial u}{\partial x}\cdot\frac{\partial u}{\partial z}+\frac{\partial v}{\partial x}\cdot\frac{\partial v}{\partial z}+\frac{\partial w}{\partial x}\cdot\frac{\partial w}{\partial z}
Angular Deformation – YZ Direction
\gamma_{yz}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\cdot\frac{\partial u}{\partial z}+\frac{\partial v}{\partial y}\cdot\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\cdot\frac{\partial w}{\partial z}
In this blog post, we will elucidate the process employed to derive these expressions for large deformations, that is, the Green deformations.
Developing expressions for Large Deformations
The latter 6 expressions allow for the calculation of deformations that occur within the domain of a continuum, or more specifically, in the context of finite element analysis, the deformation field within an element’s domain. They apply to the most general conditions of triaxial stress states, translated through solid elements, as studied in linear analysis, where these deformations only exhibited the linear terms shown in the aforementioned equations. In nonlinear analyses with large deformations, nonlinear terms are present.
The issue is that the deformations calculated by these expressions depend, in the most general case, on the position of the point around which they are calculated. That is, within a finite element, these deformations vary point by point in the most general case and are expressed by functions, which in this case are calculated from the derivatives of the functions describing the displacements within the element. The investigation of the mathematical expressions of these functions as we consider different points, or in other words, how deformations vary within the element’s domain, can be carried out through a well-known mathematical tool: the Taylor series. Thus, as deformations vary with x, y, and z, in the most general case, it is important to represent this variation point by point.
The Concept of Taylor Series
The general idea of knowing the value of a function for a given value of the variable or independent variables, based on the knowledge of the function’s value at a previous value of that variable, can be carried out, as we said, with the help of the so-called Taylor series, which expresses a function as a power series.
Taylor Series for One Variable
For example, for a function of one variable x, the value of the function f(x) is given by:
{f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}{(x-a)}^2+\frac{f'''(a)}{3!}{(x-a)}^3+…+\frac{f^n(a)}{n!}{(x-a)}^n}
where f'(a), f''(a) and f'''(a) ,…, f^n(a) are, respectively, the first, second, third, …, nth derivatives of the function f(x) at the point x = a.
Large Deformations Analysis
When a structure deforms, its points move from an initial state to a final state, passing through various intermediate situations. This evolution of the position of all points in the structure, as we mentioned, can be expressed through a function. It is much more efficient to describe the displacement field through its displacement components. Thus, as a given point moves because the structure deforms, the displacement components of that point also change. That is, not only the magnitude or the displacement vector constitutes a function; its displacement components are also functions. The derivatives of the functions u, v, and w that represent the displacement components allow us to obtain the deformations. These functions can be represented by the Taylor series. The higher-order terms of the series can describe the nonlinear behavior of displacements and deformations obtained through derivatives. And that’s what we’ll develop from now on. Figure 3.33 represents a structure in its initial and subsequent final states. Let’s try to describe this modification by translating the alteration of u, v, and w for the points of the structure. Let’s focus on one of them.
Displacement Component Analysis
The displacement component in the x-direction at point C can be related to the same component at point A through Taylor series, providing an expansion of a function in the vicinity of a point. In other words, we are attempting to describe, through a function, how the displacement components in the x-direction are described as a function of the position they occupy in space, relating them to neighboring points.
{u_C=u_A+{\left[\frac{\partial u}{\partial x}\right]}_A(\triangle x)+\frac12{\left[\frac{\partial^2u}{\partial x^2}\right]}_A{(\triangle x)}^2+\frac16{\left[\frac{\partial^3u}{\partial x^3}\right]}_A{(\triangle x)}^3+…} (Equation 1)
The figure above represents the segment AC, which is the x-component of segment AB, before deformation occurs, and subsequently, in the condition where segment AB transforms into A’B’. Thus, the original segment AC transforms into A’C’.
All points of this original segment AC will move until they reach A’C’. Each of these points, as they move, has components in x, y, and z. These displacement components are different for each point between A and C.
The graphs shown in the figures below represent the functions indicating the variation of displacements u, v, and w, respectively, as a function of the position of the point along segment AC. These curves are depicted for illustrative purposes and are arbitrary in nature.
From Equation 1, we have:
u_C-u_A={\left[\frac{\partial u}{\partial x}\right]}_A(\triangle x)+\frac12{\left[\frac{\partial^2u}{\partial x^2}\right]}_A{(\triangle x)}^2+\frac16{\left[\frac{\partial^3u}{\partial x^3}\right]}_A{(\triangle x)}^3+…
In particular, in the vicinity of point A, since \triangle x is very small, the values of \triangle x^2 and \triangle x^3 are negligible compared to \triangle x . Thus:
u_C-u_A={\left[\frac{\partial u}{\partial x}\right]}_A(\triangle x)
And, similarly, if we construct the same reasoning for the other functions, we will have:
v_C-v_A={\left[\frac{\partial v}{\partial y}\right]}_A(\triangle x) and w_C-w_A={\left[\frac{\partial w}{\partial z}\right]}_A(\triangle x)
The new length A’C’ will be given by:
{(A'C')}^2={\lbrack\triangle x+u_A-u_C\rbrack}^2+(v_C-v_A)^2+{(w_C-w_A)}^2
{(A'C')}^2=\lbrack{(\triangle x)}^2+2\triangle x(u_A-u_C)+{(u_A-u_C)}^2\rbrack+{(v_C-v_A)}^2+{(w_C-w_A)}^2
{(A'C')}^2=\left[\left(\triangle x\right)^2+2\triangle x\left(\frac{\partial u}{\partial x}\right)\triangle x+\left(\frac{\partial u}{\partial x}\right)^2\left(\triangle x\right)^2\right]+\left(\frac{\partial v}{\partial x}\right)^2\left(\triangle x\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\left(\triangle x\right)^2
{(A'C')}^2=\left(\triangle x\right)^2\left[1+2\frac{\partial u}{\partial x}+\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\right]
or
{(A'C')}^2-\left(\triangle x\right)^2=\left(\triangle x\right)^2\left[1+2\frac{\partial u}{\partial x}+\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\right]-\left(\triangle x\right)^2
{(A'C')}^2-\left(\triangle x\right)^2=\left(\left[1+2\frac{\partial u}{\partial x}+\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\right]-1\right)\left(\triangle x\right)^2
{(A'C')}^2-\left(\triangle x\right)^2=\left[2\frac{\partial u}{\partial x}+\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\right]\left(\triangle x\right)^2
if:
m=2\frac{\partial u}{\partial x}+\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2
A'C'=L'_1
\triangle x=L_{01}
{(A'C')}^2-{(\triangle x)}^2={(L'_1)}^2-{(L_{01})}^2=(L'_1+L_{01})(L'_1-L_{01})=(m){(L_{01})}^2
If the deformations are not too large:
L'_1+L_{01}\approx2\cdot L_{01}
\triangle L_1=(L'_1-L_{01})
Then:
2\cdot L_{01}\cdot\triangle L_1=\left(2\frac{\partial u}{\partial x}+\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\right)\cdot L_{01}^2
2\cdot\frac{L_{01}}{L_{01}}\cdot\frac{\triangle L_1}{L_{01}}=\left(2\frac{\partial u}{\partial x}+\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\right)
And we know that:
\varepsilon_x=\;\frac{\triangle L_1}{L_{01}}
\boxed{\varepsilon_x=\frac{\partial u}{\partial x}+\frac12\cdot\left[\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial w}{\partial x}\right)^2\right]}
The same process developed for the deformation \varepsilon_x can be applied to the deformations \varepsilon_y and \varepsilon_z , obtaining:
\boxed{\varepsilon_y=\frac{\partial v}{\partial y}+\frac12\cdot\left[\left(\frac{\partial u}{\partial y}\right)^2+\left(\frac{\partial v}{\partial y}\right)^2+\left(\frac{\partial w}{\partial y}\right)^2\right]}
\boxed{\varepsilon_z=\frac{\partial w}{\partial z}+\frac12\cdot\left[\left(\frac{\partial u}{\partial z}\right)^2+\left(\frac{\partial v}{\partial z}\right)^2+\left(\frac{\partial w}{\partial z}\right)^2\right]}
These expressions calculate strain for significant deformations, albeit within certain limits. Specifically, they are named as Green Deformations.
Conclusion
The development of the expressions for the angular (shear) deformations follows the same idea as the development of linear deformations. One should apply geometric relationships, observing the deformed configuration in relation to the undeformed configuration, identifying the first and second-order terms. This development is not carried out in this text, as the central idea of comparing undeformed and deformed geometry has already been applied, as well as the Taylor series technique.
References
- Elementos Finitos, A Base da Tecnologia, Análise Não Linear – Avelino Alves Filho
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