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Deformation models

A deformation model decides which changes of shape matter and how internal forces respond. A single spring represents one extension coordinate. A rod model represents axial strain, a beam model represents bending, and a continuum model represents displacement throughout a volume. More detail is useful only when its parameters and numerical resolution are physically calibrated.

Reference and current configurations

Material deformation is measured relative to a reference configuration. A material point labeled by reference position \(\mathbf X\) moves to current position

\[\mathbf x(\mathbf X,t)=\mathbf X+\mathbf u(\mathbf X,t)\]

where \(\mathbf u\) is the displacement. The label follows the material; it is not merely a fixed point in space. Reference length, area, volume, and density must match the stress and strain convention.

A rigid translation has nonzero displacement but no strain. A rigid rotation also changes positions without changing distances. A useful strain measure must distinguish shape change from rigid motion.

One axial element

For two nodes separated by current vector

\[\mathbf d=\mathbf x_j-\mathbf x_i\]

current length is \(\ell=|\mathbf d|\). With rest length \(\ell_0\), extension is \(\Delta\ell=\ell-\ell_0\). An ideal linear spring applies

\[\mathbf F_{i\leftarrow j}=k(\ell-\ell_0)\frac{\mathbf d}{\ell}\]

and the opposite force on node \(j\). This sign convention makes a stretched spring pull its endpoints together. The expression is undefined at zero current length, so collapsed elements require rejection, regularization, or a different model.

import numpy as np

first, second = np.array([0.0, 0.0]), np.array([1.1, 0.0])
stiffness, rest_length = 100.0, 1.0
separation = second - first
length = np.linalg.norm(separation)
force_first = stiffness * (length - rest_length) * separation / length
force_second = -force_first
print("Endpoint forces (N):", force_first, force_second)

The pair symmetry preserves system linear momentum when only internal spring forces act.

Relating a spring to a bar

A uniform axial bar of length \(L\), area \(A\), and Young's modulus \(E\) has small-deformation stiffness

\[k=\frac{EA}{L}\]

This relation lets a discrete axial element reproduce the bar's force-extension response. If one bar is split into \(n\) equal elements, each element length becomes \(L/n\) and its stiffness becomes \(nEA/L\). Keeping the original spring constant for every segment makes the refined chain too soft.

Mesh refinement must preserve physical material properties, not merely add nodes.

Small-strain continuum model

For small displacement gradients, infinitesimal strain tensor is

\[\boldsymbol\varepsilon=\frac12\left(\nabla\mathbf u+(\nabla\mathbf u)^{\mathsf T}\right)\]

The symmetric part removes infinitesimal rigid rotation. Linear elasticity relates stress and strain through a material stiffness tensor. For a homogeneous isotropic material,

\[\boldsymbol\sigma=\lambda\,\operatorname{tr}(\boldsymbol\varepsilon)\mathbf I+2G\boldsymbol\varepsilon\]

where \(\lambda\) and \(G\) are elastic constants. This is a small-strain constitutive law, not a formula for arbitrary large deformation.

Large deformation

At large rotation or stretch, infinitesimal strain can report spurious strain for a rigid rotation. Finite-deformation models use measures derived from the deformation gradient

\[\mathbf F=\frac{\partial\mathbf x}{\partial\mathbf X}\]

and pair them with compatible stress measures and energy functions. The symbol \(\mathbf F\) here is a tensor and must not be confused with a force vector.

Objectivity requires stored energy to remain unchanged under a superposed rigid rotation. A material law that produces elastic energy merely because the whole body rotates is incorrect.

Rods, beams, membranes, and solids

A rod model focuses on axial stretch and possibly twist. A beam model adds bending and often shear while assuming cross-sectional behavior. A membrane resists in-plane stretch but has little or no bending stiffness unless a shell model adds it. A solid continuum represents three-dimensional stress.

Choose based on the question. A chain of axial springs can bend freely unless geometry or additional bending terms resist changes of angle. Making a cloth image visually smooth does not supply physical bending stiffness. Likewise, a two-dimensional mesh may not represent thickness stress or out-of-plane buckling correctly.

Bending model

For a slender Euler-Bernoulli beam under small deflection, curvature is approximately the second derivative of transverse displacement \(w(x)\). Bending moment and curvature relate as

\[M_b=EI\frac{d^2w}{dx^2}\]

where \(I\) is geometric, not mass moment of inertia. Both use length to the fourth or mass-length-squared contexts differently; naming the quantity prevents confusion.

The Euler-Bernoulli assumptions can fail for deep beams, large deflection, shear-dominant response, anisotropy, or dynamic wavelengths comparable to cross-section size.

Damping and time dependence

Elastic force depends on configuration. Viscous damping depends on rate and dissipates energy. For an axial element, a simple dashpot force along the element can depend on relative axial velocity. It should vanish under equal rigid translation of both nodes.

Too much numerical damping can make an inaccurate method look stable. Physical damping parameters should come from material or system behavior, while algorithmic dissipation should be documented separately.

Viscoelastic materials depend on loading history. Plastic materials retain permanent strain after unloading. Damage and fracture change stiffness or connectivity. None can be represented by a single-valued force of current extension alone; they require internal state.

Constraints and boundary conditions

Displacement boundary conditions prescribe motion. Force boundary conditions prescribe tractions or loads. A fixed node represents a support that can exchange momentum and energy with the deformable body. Its reaction should be calculated or reported rather than erased by overwriting motion invisibly.

Insufficient constraints leave rigid-body zero modes and a singular static stiffness matrix. Excess or incompatible constraints can create artificial stress. Test a free body for zero energy under rigid translation and rotation, then test supported cases.

Discrete networks

For a spring network, visit each edge once and add opposite endpoint forces from the same state snapshot. Advance all free nodes only after all forces are assembled. Updating one node during force assembly makes results depend on storage order.

positions = np.array([[0.0, 0.0], [1.1, 0.0], [2.0, 0.0]])
edges = [(0, 1), (1, 2)]
forces = np.zeros_like(positions)
for i, j in edges:
    separation = positions[j] - positions[i]
    length = np.linalg.norm(separation)
    force = stiffness * (length - rest_length) * separation / length
    forces[i] += force
    forces[j] -= force
assert np.allclose(forces.sum(axis=0), 0.0)
print("Node forces (N):", forces)

The total internal force should be zero to numerical precision. For central edge forces, internal torque should also cancel about a common origin.

Calibration under refinement

Adding elements changes element size and often changes stable time steps. Preserve total mass, density, dimensions, and continuum modulus. Compare the same physical observable— such as endpoint displacement under a fixed load—at successive resolutions.

A lumped-mass choice assigns element mass to nodes. A consistent mass matrix distributes inertia differently. Both can converge, but their dynamic frequencies and numerical properties differ at coarse resolution. Record the choice.

Energy and validation

For conservative elements, total internal force should equal the negative gradient of stored energy. Check this relation by finite differences. A rest configuration with no external load should have zero elastic force. Rigid translations and rotations should not change objective elastic energy.

Track kinetic energy, elastic energy, external work, damping loss, and constraint work. Reduce both spatial resolution size and time step independently. A visually smooth shape can coexist with wrong stiffness, energy, or stress.

Reliability checks

Deformation models range from one spring coordinate to tensor fields. Their fidelity comes from choosing the right kinematics and constitutive law, preserving material properties under discretization, and validating forces and energy independently of appearance.

A worked Python notebook

A spring element has rest length one metre and stiffness 100 newtons per metre. The question for this calculation is precise: how does axial tension change when we change element length, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Negative tension denotes compression in this ideal element; buckling is not included. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.8 to 1.2 m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for element length, expressed in m. The output is axial tension, expressed in N. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 100 q - 100\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1.1 m, the expression gives 10 N. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("100*(q-1)", locals={"q": q})
reference_input = 1.1
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted axial tension:", reference_output, "N")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives -20 N. The midpoint, at 1 m, gives 0 N. The upper endpoint gives 20 N. Read these together with the assumptions: they describe different choices of element length within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.8, 1.0, 1.2])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m -> {output_value:.6g} N")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows element length; the vertical axis shows axial tension. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.8, 1.2, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Element length (m)")
ax.set_ylabel("Axial tension (N)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does axial tension change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1.1 m is 100 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 100\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.102
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "N")
print("Full expression:", full_prediction, "N")
print("Approximation difference:", full_prediction - linear_prediction, "N")

Here the changed input is 1.102 m. The tangent estimate is 10.2 N, while the full expression gives 10.2 N. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 0.3999999999999999
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of element length inside the stated interval, predict whether axial tension will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Negative tension denotes compression in this ideal element; buckling is not included. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of N: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(1.2) - evaluate(0.8))
for points in [101, 201, 401]:
    grid = np.linspace(0.8, 1.2, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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